Abstract: Systems and methods are provided for measuring aberration in a lithographic apparatus. A radiation beam is modulated using an array of individually controllable elements, and the modulated beam is projected using a projection system. A pattern is provided on the array of individually controllable elements to modulate the radiation beam, and the pattern comprises a repeating structure that is formed from a plurality of features that are dimensioned such that first order diffraction of the radiation beam substantially fills the pupil of the projection system. A sensor detects the projected radiation and measures interference in the radiation projected by the projection system. Aberration in the detected radiation beam is then measured.

Abstract: Multi-exposure lithography systems are provided for improved overlay accuracy. In one aspect, a method for multi-exposure lithography operates by determining overlay parameters corresponding to each of a plurality of sub-layouts, inputting the overlay parameters into an exposure system, exposing each sub-layout to photoresist on a wafer by using the exposure system, wherein prior to the exposure process for a given sub-layout, a correction process is performed for the sub-layout using a corresponding overlay parameter to correct an overlay of the sub-layout, and developing the exposed photoresist after exposing all of the sub-layouts.

Abstract: One or more patterning arrays are mounted to a mounting plate via height adjustment structures that enable the flatness of the active surfaces of the patterning arrays to be controlled. The height adjustment structures may comprise an array of piezoelectric actuators or screws. Alternatively, the backside of the patterning means may be polished to optical flatness and bonded by crystal bonding to an optically flat surface of a rigid mounting body.

Abstract: Devices and methods are disclosed for holding an object, particularly a planar object. An exemplary device has a chuck and pressure-changing device. The chuck has an object-mounting surface and a deformable membrane coupled to the object-mounting surface such that conformational changes in the membrane produce corresponding changes in the object-mounting surface. The chuck has a first cavity separated by the membrane from the chuck cavity. The pressure-changing device is coupled to the first cavity to change pressure in the first cavity, relative to outside it, sufficiently to produce a conformational change of the membrane and a corresponding change in the object-mounting surface sufficient to reduce the force with which the object is being held to the object-mounting surface. The pressure change can be a pressure increase or decrease.

Abstract: A lithographic method is provided that includes using an illumination system to provide a beam of radiation having an illumination mode, using a patterning device to impart the radiation beam with a pattern in its cross-section, and projecting the patterned radiation beam onto a substrate. The illumination mode is adjusted after the radiation beam has been projected onto the substrate. The adjustment is arranged to reduce the effect of optical aberrations due to lens heating on the projected pattern during projection of the pattern onto a subsequent substrate.

Abstract: For the correction of anamorphism in the case of a projection lens of an EUV projection exposure apparatus for wafers it is proposed to tilt the reticle bearing the pattern to be projected and preferably also the wafer by a small angle about an axis that is perpendicular to the axis A of the lens and perpendicular to the scan direction and that in each instance passes through the middle of the light field generated on the reticle or on the wafer. For the correction of a substantially antisymmetric quadratic distortion the reticle and/or the substrate is instead rotated about an axis of rotation that is disposed at least approximately parallel to an optical axis of the projection lens.

Abstract: A digital exposure apparatus for forming a pattern on a topside surface of a substrate has an alignment unit, which detects a position of the substrate from image data obtained through a camera from a reference mark provided on the topside surface of the substrate. The alignment unit further has a Z-direction sensor to measure a fluctuation amount ? of the topside surface of the substrate from a predetermined focal plane of the camera. Depending upon the measured amount ?, a set of distortion correction data is selected from or calculated on the basis of previously stored distortion correction data. The image data of the reference mark is corrected with the selected or calculated distortion correction data, so that errors induced by the fluctuation from the focal plane are corrected without adjusting the position of the substrate in the direction of an optical axis of the camera of the alignment unit.

Abstract: A projection exposure mask acceptance decision system includes assurance object measuring unit to measure quality assurance objects relating to projection exposure mask, first exposure characteristic deterioration quantity calculating unit to calculate first exposure characteristic deterioration quantity caused by deviations in average values of the quality assurance objects measured by the measuring unit, second exposure characteristic deterioration quantity calculating unit to calculate second exposure characteristic deterioration quantity caused by dispersion in the quality assurance objects measured by the measuring unit, sum calculating unit to calculate simple sum of the first and second quantity, root sum square calculating unit to calculate root sum square of the first and second quantity, entire exposure characteristic deterioration quantity calculating unit to calculate entire exposure characteristic deterioration quantity as an interior division value of the simple sum and root sum square, and judg

Abstract: A method for determining rotational error portion of total misalignment error in a stepper. In one embodiment, the method comprises a series of steps in a stepper, starting with the step of receiving a wafer, having a first pattern and an error-free fine alignment target, in the stepper. In another step, the wafer is aligned in the stepper using the error-free fine alignment target. Then a second pattern is created on the wafer overlaying said first pattern. In another step, the rotational error portion of the total misalignment error is determined by measuring the circumferential misalignment between the first pattern and the second pattern.

Abstract: A method of focus variation is described herein to achieve a one-step exposure of a wafer without the limitation of applying a complex y-tilt to a wafer stage. The position of the wafer surface to be exposed is periodically varied with respect to the focal plane, or vice versa. This relative movement between the focal plane, or best focus position along the optical axis and the wafer stage, or the wafer surface, is achieved by applying a movement to at least one of the reticle stage, one or more of the optical elements of the projection lens, and the wafer stage. The frequency of the movement is selected in dependence of the laser frequency (upper limit) or the scanning frequency (lower limit).

Abstract: In the case where the previous process (X) and the previous process (Y) are different in step 310, only a distortion amount in an X-axis direction is extracted from image distortion data of the previous process (X) in Step 316 and only a distortion amount in a Y-axis direction is extracted from image distortion data of the previous process (Y) in Step 318, and then in Step 320, image distortion data is created by synthesizing the extracted distortion amounts, and the synthesized image distortion data is used for subsequent adjustment of projected images. With this operation, the distortion of projected images can be adjusted per axis and accordingly overlay exposure with high accuracy can be realized.

Abstract: A lithographic mask is illuminated with light from different directions such that intensities of a plurality of incident beams of light provide a largest possible integrated process window defined in terms of an allowed range for defining shapes. Constrained sets of intensity parameters are imposed. A first set of intensity parameters represents maximum possible intensities that can be permitted for overexposed tolerance positions. A second set of intensity parameters represents minimum possible intensities that can be permitted for underexposed tolerance positions. Optimum source intensities of incident beams are defined using a linear program and constraints. The optimum source intensities maximize an integrated range of dose and focal variations without causing printed shapes to depart from the allowed range. Apparatus are detailed and variations are described.

Abstract: A lithographic apparatus according to one embodiment includes an alignment system for aligning a substrate. The alignment system comprises an illuminator system configured to illuminate an alignment mark on the substrate with an illumination spot, the alignment mark comprising a plurality of lines and spaces. The system also includes a combiner system configured to transfer two images of the illuminated alignment mark without spatial filtering of the images, rotate the images 180° relatively to each other, and combine the two images; and a detection system configured to detect an alignment signal from the combined images and to determine a unique alignment position by selecting a specific one of extreme values in the detected alignment signal.

Abstract: An automatic correction method for a direct exposure apparatus illuminates two exposure elements, which are included in adjacent exposure heads separately and which are to expose an identical line on an exposure target, among exposure elements arranged in a two-dimensional manner with an inclination of relative movement of an exposure target in a plurality of exposure heads arranged in the direction of the relative movement of the exposure target; detects the illumination of the two exposure elements by using a sensor board (11) that is moved, on the side where the exposure heads are illuminated, in the direction in which the exposure heads are arranged; and calculates correction amounts for correcting so that the two exposure elements can expose the identical line mentioned above, based on a detection result of the illumination of the exposure elements by the sensor board (11).

Abstract: A device and a method for testing an exposure apparatus is disclosed. A testing device includes a substrate, and a plurality of block patterns, each of which has a top area varying with an area of a shot region of the exposure apparatus, having at least two different heights located on the substrate. Additionally, the method for testing an exposure apparatus includes using the exposure apparatus to perform an exposure process on the testing device or on the testing device having a photoresist layer thereon, and testing the performance of the exposure apparatus through comparing surface information of the testing device computed by the exposure apparatus with actual surface information of the testing device or through examining photoresist patterns formed on the testing device.

Abstract: A lithographic apparatus is disclosed having a deformable lens element through which a patterned radiation beam is arranged to pass before reaching a substrate and having a deformable lens actuator configured to transmit a combination of a force substantially parallel to the optical axis of the projection system and a localized torque about an axis substantially perpendicular to the optical axis independently at a plurality of sub-regions on the deformable lens element.

Abstract: Disclosed is an exposure method which illuminates a first object with an exposure beam and exposes a second object with the exposure beam through the first object and a projection optical system, wherein at least a part of one of the first object and the projection optical system is irradiated with a light beam having a wavelength range different from that of the exposure beam through a space waveguide mechanism, to correct an imaging characteristic of the projection optical system.

Abstract: A device manufacturing method includes exposing a substrate with a patterned beam of radiation formed by a reticle mounted on a displaceable reticle stage, determining a non-linear function for approximating a height and a tilt profile of a surface of the reticle with respect to the reticle stage, and controlling a displacement of the reticle stage during exposure of the substrate in accordance with the non-linear function.

Abstract: An exposure apparatus illuminates a pattern on an original form, introduces light from the pattern to a plate, and exposes the plate. The exposure apparatus includes at least one optical element, and forcing member that applies a force to the at least one optical element in a non-contact manner.

Abstract: A drive unit drives a wafer stage in a Y-axis direction based on a measurement value of an encoder that measures position information of the wafer stage in the Y-axis direction and based on information on the flatness of a scale that is measured by the encoder. In this case, the drive unit can drive the wafer stage in a predetermined direction based on a measurement value after correction in which a measurement error caused by the flatness of the scale included in the measurement value of the encoder is corrected based on the information on the flatness of the scale. Accordingly, the wafer stage can be driven with high accuracy in a predetermined direction using the encoder, without being affected by the unevenness of the scale.

Abstract: An exposure apparatus for exposing a substrate. A stage holds and moves the substrate, a measuring device measures a position of a surface of the substrate, and a controller defines an arrangement of measurement points with respect to each of a plurality of shot regions arranged along a straight line, and causes the measuring device to sequentially measure positions of the surface with respect to the defined measurement points in the plurality of shot regions, while the stage moves the substrate along the straight line. A timing at which the measuring device measures a position of the surface is controlled so that a first time interval and a second time interval are different from each other.

Abstract: A second communications server obtains adjustment information of an adjustment unit in an exposure apparatus and information related to image forming quality (such as wavefront aberration) of a projection optical system under an exposure condition that serves as a reference, via a first communications server, and then calculates an optimal adjustment amount of the adjustment unit under a target exposure condition based on the information. In addition, the second communications server obtains adjustment information of the adjustment and actual measurement data of image forming quality of the projection optical system via the first communications server, and then calculates the optimal adjustment amount of the adjustment unit under the target exposure condition based on the information. And, the second communications server controls the adjustment unit in the exposure apparatus via the first communications server, based on the calculations results.

Abstract: An exposure system for manufacturing flat panel displays (FPDs) includes a reticle stage and a substrate stage. A magnification reflective optical system images the reticle onto the substrate. The system may be a 2× magnification system, or another magnification that is compatible with currently available mask sizes. By writing reticles with circuit pattern dimensions that are one-half the desired size for an FPD, a 2× optical system can be used to expose FPDs. The designs for the 1.5× and larger magnification optical systems all typically have at least three powered mirrors. A corrector, positioned either near the reticle or near the substrate, can be added to the three mirror design to improve the systems optical performance. The corrector may be a reflective, or a refractive design. The corrector can have an aspheric surface, and optionally a powered surface.

Abstract: A lithographic projection apparatus is disclosed that includes a predictive system configured to predict changes in projection system aberrations with time with respect to measured aberration values, a modelling system configured to determine an application-specific effect of said predicted projection system aberration changes on at least one parameter of an image for a selected pattern, a control system configured to generate a control signal specific to the selected pattern according to said predicted projection system aberration changes and their application-specific effect on the at least one parameter of the image, and an image adjusting system, responsive to the control signal, to compensate for the application-specific effect of said predicted projection system aberration changes on the image. Adjustments may therefore be determined optimally for a given application.

Abstract: A method of optimizing adjustable settings of adjustable elements of a projection system in a lithographic apparatus is disclosed that includes determining an object spectrum for a pattern and an illumination arrangement, determining a symmetry of the object spectrum, constructing a merit function for a wavefront in a pupil plane of the projection system with the settings of the adjustable elements as variables with reference to the symmetry, and optimizing the merit function.

Abstract: A lithographic apparatus includes an illumination system configured to condition a radiation beam, a support for a patterning device, a substrate table for a substrate, a projection system, and a control system. The patterning device is capable of imparting the radiation beam with a pattern in its cross-section to form a patterned radiation beam. The projection system is configured to project the patterned radiation beam as an image onto a target portion of the substrate along a scan path. The scan path is defined by a trajectory in a scanning direction of an exposure field of the lithographic apparatus. The control system is coupled to the support, the substrate table and the projection system for controlling an action of the support, the substrate table and the projection system, respectively. The control system is configured to correct a local distortion of the image in a region along the scan path by a temporal adjustment of the image in that region.

Abstract: A lithographic apparatus includes an illumination system for providing a beam of radiation, and a support structure for supporting a patterning device. The patterning device serves to impart the projection beam with a pattern in its cross-section. The apparatus also includes a substrate table for holding a substrate, and a projection system for projecting the patterned beam onto a target portion of the substrate. At least one of the patterning device, or the projection system, and the illumination system includes a reflector assembly that includes a reflector substrate with a reflective surface for reflecting part of incident radiation, and a heat exchanger system that is constructed and arranged to exchange heat with the reflector substrate. The heat exchanger system includes a thermally active element that is disposed in a recess of the reflector substrate at a side of the reflector substrate that is different from the reflective surface.

Abstract: In a lithographic apparatus, a corrective irradiation procedure is performed using an illumination mode arranged so as to heat a selected part of an element of the projection system near a pupil plane thereof that is relatively unheated during production exposure. The corrective irradiation procedure aims to improve uniformity of optical element heating of the projective system and/or to reduce a phase gradient.

Abstract: A second communications server obtains adjustment information of an adjustment unit in an exposure apparatus and information related to image forming quality (such as wavefront aberration) of a projection optical system under an exposure condition that serves as a reference, via a first communications server, and then calculates an optimal adjustment amount of the adjustment unit under a target exposure condition based on the information. In addition, the second communications server obtains adjustment information of the adjustment and actual measurement data of image forming quality of the projection optical system via the first communications server, and then calculates the optimal adjustment amount of the adjustment unit under the target exposure condition based on the information. And, the second communications server controls the adjustment unit in the exposure apparatus via the first communications server, based on the calculations results.

Abstract: A pattern forming method is proposed for easy correction of a pattern-size variation occurring in an etching process. An energy beam is radiated onto a resist-applied target while the energy beam is adjusted to correct the pattern-size variation occurring in the etching process. The resist on the target is developed to form a resist pattern. The target is etched with the resist pattern as a mask, thus forming patterns thereon.

Abstract: A radiation distribution measurement system for measuring a phase distribution of a beam of radiation and/or a pupil distribution of a projection system includes a transparent carrier plate, a grating and/or a pinhole configured at a first side of the transparent carrier plate, and a camera at an opposite side of the transparent carrier plate. The measurement system also includes a radiation filter between the transparent carrier plate and the camera, with a transmissivity that is lowest at the center of the filter and gradually and concentrically increases towards the outside of the filter. By placing the filter with its specific transmissivity, the difference in intensity across the wave front sensor 10 (i.e. the gradient in intensity) is compensated. The intensity of the light incident on the camera is made more uniform resulting in an improved performance of the measurement system.

Abstract: A thick pellicle is allowed to have a non-flat shape and its shape is characterized to calculate corrections to be applied in exposures to compensate for the optical effects of the pellicle. The pellicle may be mounted so as to adopt a one-dimensional shape under the influence of gravity to make the compensation easier.

Abstract: A projection objective of a microlithographic projection exposure apparatus contains a plurality of optical elements arranged in N?2 successive sections A1 to AN of the projection objective which are separated from one another by pupil planes or intermediate image planes. According to the invention, in order to correct a wavefront deformation, at least two optical elements each have an optically active surface locally reprocessed aspherically. A first optical element is in this case arranged in one section Aj, j=1 . . . N and a second optical element is arranged in another section Ak, k=1 . . . N, the magnitude difference |k?j| being an odd number.

Abstract: An exposure apparatus and method in which an image is formed on a recording medium mounted on a recording stage by irradiating a light beam based on image data from a recording head while the recording head and the recording stage are relatively moved, in the exposure apparatus and method, storing in advance, in a storage section, shift-amount data of a detected displacement of an image-form position with respect to a stage surface of the recording stage in a direction intersecting a direction in which the recording stage moves, which displacement occurs accompanied by movement of the recording stage; carrying out shifting for each of pixels of an image formed by the image data based on the shift-amount data stored in the storage section; and controlling exposure for the recording medium based on image data of the shifted image, is provided to reduce distortion of the image.

Abstract: For the correction of anamorphism in the case of a projection lens of an EUV projection exposure apparatus for wafers it is proposed to tilt the reticle bearing the pattern to be projected and preferably also the wafer by a small angle about an axis that is perpendicular to the axis A of the lens and perpendicular to the scan direction and that in each instance passes through the middle of the light field generated on the reticle or on the wafer. For the correction of a substantially antisymmetric quadratic distortion the reticle and/or the substrate is instead rotated about an axis of rotation that is disposed at least approximately parallel to an optical axis of the projection lens.

Abstract: A projection optical system includes an optical element that includes and locally uses a reflective or refractive area that is substantially axially symmetrical around an optical axis, the optical element being rotatable around the optical axis.

Abstract: The invention relates to a method for a focus test. The method includes performing a first projection by using a radiation beam to project a first reference mark onto a substrate to generate a first reference mark image, and performing a second projection by using a radiation beam to project a first sample mark onto the substrate to generate a first sample mark image, wherein the first reference mark image and the first sample mark image at least partially overlap and the second projection is relatively focus sensitive compared to the first projection.

Abstract: A lithographic apparatus is disclosed that includes a support constructed to support a patterning device, the patterning device being capable of imparting a radiation beam with a pattern in its cross-section to form a patterned radiation beam and a substrate table constructed to hold a substrate. Further, the lithographic apparatus includes a projection system configured to project the patterned radiation beam onto a target portion of the substrate, the projection system being mounted to a reference element of the lithographic apparatus by a resilient mount to reduce a transfer of high frequency vibration from the reference element to the projection system and a control system to counteract a position error of the substrate table and the support relative to the projection system.

Abstract: Model Based Optical Proximity Correction (MOPC) biasing techniques may be utilized for optimizing a mask pattern. However, conventional MOPC techniques do not account for influence from neighboring features on a mask. This influence may be factored in the following manner—first, generating a predicted pattern from a target pattern and selecting a plurality of evaluation points at which biasing may be determined. Next, a set of multivariable equations are generated for each evaluation point, each equation representing influence of neighboring features on a mask. The equations are solved to determine that amount of bias at each evaluation point, and the mask is optimized accordingly. This process may be repeated until the mask pattern is further optimized.

Abstract: An exposure apparatus according to one aspect of the present invention includes an illumination optical system for illuminating, in a slit-shaped illumination area, a pattern of a reticle with a light from a light source, a projection optical system for projecting an image of the pattern onto a wafer, the projection optical system including a mirror, a detection system for detecting a positional offset of a light spot while the wafer is exposed with the image, and a drive unit for driving an optical element of the projection optical system, while the wafer is exposed with the image based on the positional offset detected by the detection system, wherein the detection system includes a light source for emitting a detection light beam, and a light-receiving device for receiving the light spot of the detection light beam via the projection optical system, and wherein the light source is located at a position substantially equivalent to the wafer, the light-receiving device is located at a position substantially

Abstract: Aberrations in an optical system can be detected and measured using a method comprised of a test target in the object plane of a projection system and imaging a photoresist film with the system. The test target comprises at least one open figure which comprises a multiple component array of phase zones, where the multiple zones are arranged within the open figure so that their response to lens aberration is interrelated and the zones respond uniquely to specific aberrations depending on their location within the figure. This is a unique and new method of detecting a variety of aberration types including coma, spherical, astigmatism, and three-point through the exposure of a photoresist material placed in the image plane of the system and the evaluation of these images.

Abstract: An exposure apparatus includes a projection optical system, an original stage having a first reference mark, a substrate stage, and a measurement instrument configured to measure first image properties of a mark formed on the original with the projection optical system, via the original and the projection optical system. The measurement instrument is also configured to measure the properties of a second image with the projection optical system of the first reference mark, via the reference mark and the projection optical system. The exposure apparatus also includes a calculating unit for calculating a first heat change coefficient due to the projection optical system and a second heat change coefficient due to the original, with relation to the properties of images formed by the projection optical system.

Abstract: An exposure apparatus which exposes a substrate to light is disclosed. The exposure apparatus for exposing a substrate to light, the apparatus comprises: a projection optical system configured to project light from a reticle onto the substrate, the projection optical system including at least one optical element driven to adjust aberration of the projection optical system; a driver configured to drive the at least one optical element; and a calculator configured to calculate a target amount to which the driver drives the at least one optical element, wherein the apparatus is configured so that the driver drives the at least one optical element a plurality of times based on outputs from the calculator in a non-exposure period during which the substrate is not exposed to light.

Abstract: The aligning of a wafer with a reticle in photolithographic equipment is carried out using a feed forward method. In the method, a wafer is loaded onto an exposure apparatus, the wafer is aligned with a reticle, the state of alignment is measured, alignment data representative of the state of alignment is produced, and a database is searched for an alignment data type under which the alignment data falls. The database may also be searched for overlay data related to the alignment data. A correction value matched to the alignment data type is obtained. The correction value maybe calculated from the overlay data. The alignment of the wafer is corrected by applying the correction value to the alignment data. Finally, the aligned wafer is exposed.

Abstract: A simulator of a lithography tool includes a correcting parameter memory storing a correcting scaling value to correct a focus error of a projection optical system in the lithography tool and a correcting bias to correct a critical dimension error generated in the lithography tool. A model simulation engine simulates an image formation under a corrected focus calculated by multiplying a defocus of the projection optical system by the correcting scaling value to model a calculated critical dimension of an image. A bias corrector adds the correcting bias to the calculated critical dimension to correct the image.

Abstract: A method of adjusting exposure error for multiple products is described. First, one Photo Feed Back System (PFBS) suited to host-product or miscellaneous product is chosen. Different standard points and compensation difference for host-product or miscellaneous product are provided. Then, the PFBS parameter is calculated as an exposure adjustment value. The standard point and compensation difference for miscellaneous product are dependent on host-product.

Abstract: An exposure apparatus includes a projection optical system for projecting an exposure pattern, onto an object to be exposed, and a measuring apparatus for measuring, as an interference fringe, optical performance of the projection optical system, wherein the measuring apparatus includes an optical element having opposing first and second surfaces, wherein the first surface has a first measurement pattern, and the second surface has a second measurement pattern and is closer to the projection optical system than the first measurement pattern, and wherein the measuring apparatus introduces light into the projection optical system via first and second measurement patterns.

Abstract: A method for evaluating a local flare in an exposure tool, includes: measuring a projection light intensity distribution by transferring a monitor mask pattern onto a semiconductor substrate; calculating a first ratio between an illumination light intensity on the monitor mask pattern and a first projection light intensity calculated based on the monitor mask pattern; calculating a distribution function of a local flare, due to a mask pattern coverage of the monitor mask pattern, based on the first ratio and the projection light intensity distribution; dividing a design mask pattern into a plurality of unit areas; calculating a second ratio between the illumination light intensity on each of the unit areas and a second projection light intensity calculated based on the design mask pattern; and calculating a local flare intensity in each of the unit areas, based on the second ratio and the distribution function.

Abstract: A lithographic apparatus comprises an illumination system having optical elements capable of conditioning a radiation beam to comprise in cross-section a first portion of linearly polarized radiation and a second portion of unpolarized or circularly polarized radiation. The apparatus further comprises a support constructed to support a patterning device is provided, the patterning device being capable of imparting the illuminating radiation beam with a pattern in its cross-section to form a patterned radiation beam. A substrate table is provided to hold a substrate, whilst a projection system is provided and configured to project the patterned radiation beam onto a target portion of the substrate.

Abstract: A projection optical system is a catoptric system in which a field of view region and an imaging region are located spaced from an optical axis, in which a numerical aperture of light reaching each point on an image plane is substantially uniform regardless of an image height and a direction. An aperture stop for defining the numerical aperture of the projection optical system is provided, and the aperture stop is provided with an aperture portion in a predetermined shape in which the numerical aperture of light reaching each point within a predetermined region is substantially uniform over the predetermined region, that is, in a shape in which dimensions concerning two directions perpendicular to each other are different from each other.