Flatness error resistant photomask measurement technique

Abstract

A method of determining flatness of a substrate includes measuring a first flatness measurement of the substrate in a first orientation relative to a vertical direction, measuring a second flatness measurement of the substrate in a second orientation relative to the vertical direction, measuring a third flatness measurement of the substrate in a third orientation relative to a vertical direction, and measuring a fourth flatness measurement of the substrate in a fourth orientation relative to a vertical direction, each of the first, second, third, and fourth orientations being in a different orientation relative to the vertical direction. The method further includes generating a first set of difference values, fitting the first set of difference values to a respective orthogonal polynomial, and generating a true flatness of the substrate.

Description

[0001] This application claims priority to U.S. Provisional Application Serial No. 63 / 600,076, filed November 17, 2023, pursuant to 35 USC §120, the contents of which are based and are incorporated herein by reference in their entirety. Technical Field

[0002] This disclosure relates generally to photomask measurement and fabrication, and more specifically, to techniques for measuring the flatness of photomasks. Background Technology

[0003] Optical lithography is a critical process for manufacturing a wide range of electronic devices, such as microelectronic devices and integrated circuits, and other examples, which are widely used in industrial, academic, and commercial applications (e.g., computers, automobiles, wearable devices such as smartwatches, mobile electronic devices such as smartphones and tablets). Optical lithography (e.g., extreme ultraviolet (EUV) lithography) can be used to transfer an image from a pattern (e.g., a master pattern) on a photomask (e.g., a photolithographic mask) to a semiconductor material (e.g., a semiconductor material such as crystalline silicon). In some cases, the semiconductor material may include a photosensitive coating, such that exposing the photosensitive coating enables one or more processes to be performed on the semiconductor material. For example, after exposing the photosensitive coating, the semiconductor material may be processed to etch the pattern from the photomask into a substrate or replace the exposed pattern with a new material, and other processes. In some examples, this process may be repeated multiple times to produce semiconductor material and ultimately an electronic device (e.g., a final product).

[0004] However, in some cases, the flatness of the photomask can affect the accuracy of the patterning transferred to the semiconductor material. For example, if the flatness of the patterned surface of the photomask deviates from the desired flatness (e.g., true flatness, ideal plane), errors may be introduced during the physical patterning of the semiconductor material, making it impossible for the semiconductor material to be patterned according to the desired image. In some such cases, the flatness of the photomask can be measured to ensure that the flatness meets the desired flatness or to ensure that flatness is identified and taken into account during the patterning of the semiconductor material so that flatness errors do not cause errors in the patterned semiconductor material. Summary of the Invention

[0005] The described techniques relate to improved methods, systems, apparatus, and devices for photomask measurement techniques that support flatness error tolerance. Generally, the described techniques involve measuring a photomask in various orientations and performing calculations to determine the flatness (e.g., true flatness) of the photomask. For example, the flatness of the photomask can be measured by one or more interferometers in one or more orientations (e.g., the photomask is positioned near a vertical position), and a controller can use the measurements to determine a measured value of the photomask flatness, such as a gravity error measurement of the photomask flatness. In some such examples, the gravity error measurement of the photomask flatness can be associated with the flatness of the photomask, which is independent of the gravity acting on the photomask (e.g., gravity-independent). The gravity error measurement of the photomask flatness can be used for precise patterning of a substrate, such that the flatness of the photomask is taken into account, and no errors are introduced on the substrate due to deviations in the flatness of the photomask (e.g., during patterning).

[0006] According to various aspects of this disclosure, a method for determining the flatness of a substrate is disclosed, the method comprising: measuring a first flatness measurement value of the substrate in a first orientation relative to a vertical direction; measuring a second flatness measurement value of the substrate in a second orientation relative to the vertical direction; measuring a third flatness measurement value of the substrate in a third orientation relative to the vertical direction; and measuring a fourth flatness measurement value of the substrate in a fourth orientation relative to the vertical direction, wherein each of the first orientation, the second orientation, the third orientation, and the fourth orientation is in a different orientation relative to the vertical direction, generating the first flatness measurement value, the second flatness measurement value, and the third flatness measurement value. The first set of differences between one of the flatness measurement values ​​and the fourth flatness measurement value and the other of the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value is fitted to a corresponding orthogonal polynomial to generate a fifth flatness measurement value. A flatness measurement value is selected from the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value. The orthogonal polynomial component associated with the selected flatness measurement value is replaced with the orthogonal polynomial component associated with the fifth flatness measurement value, and the true flatness of the substrate is generated.

[0007] According to various aspects of this disclosure, an apparatus is disclosed comprising a support configured to hold a substrate in one or more orientations using one or more support members; and an interferometer configured to measure a first flatness measurement of the substrate in a first orientation relative to a vertical direction, a second flatness measurement of the substrate in a second orientation relative to the vertical direction, a third flatness measurement of the substrate in a third orientation relative to the vertical direction, and a fourth flatness measurement of the substrate in a fourth orientation relative to the vertical direction, each of the first, second, third, and fourth orientations being in a different orientation relative to the vertical direction. The device further includes a controller configured to generate a first set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements; fit the first set of differences to a corresponding orthogonal polynomial to generate a fifth flatness measurement; select a flatness measurement from the first, second, third, and fourth flatness measurements; replace an orthogonal polynomial component associated with the selected flatness measurement with an orthogonal polynomial component associated with the fifth flatness measurement; and generate the true flatness of the substrate.

[0008] According to various aspects of this disclosure, a method for determining the flatness of a substrate is disclosed, the method comprising: measuring a first flatness measurement of the substrate in a first orientation relative to a vertical direction; measuring a second flatness measurement of the substrate in a second orientation relative to the vertical direction; generating a difference measurement between the first flatness measurement and the second flatness measurement; fitting the difference measurement to a corresponding first orthogonal polynomial; selecting a flatness measurement from the first and second flatness measurements, and fitting the selected flatness measurement to a corresponding second orthogonal polynomial; subtracting the first orthogonal polynomial from the second orthogonal polynomial to generate an error estimate; and generating the true flatness of the photomask based at least in part on the error estimate. Attached Figure Description

[0009] Figure 1 An example of a system for supporting substrate measurement techniques that resist flatness errors according to various aspects of this disclosure is shown.

[0010] Figure 2A , 2BFigures 2C and 2D illustrate examples of substrate orientation diagrams that support substrate measurement techniques for resisting flatness errors according to various aspects of this disclosure.

[0011] Figure 3A An example of a process flow for a substrate measurement technique supporting anti-flatness error according to various aspects of this disclosure is shown.

[0012] Figure 3B and 3C Each illustrates an exemplary mapping of flatness measurements according to various aspects of this disclosure.

[0013] Figure 4 An example of a process flow for a substrate measurement technique supporting anti-flatness error according to various aspects of this disclosure is shown.

[0014] Figure 5 A block diagram of a substrate measurement system supporting photomask measurement technology for flatness error according to various aspects of this disclosure is shown.

[0015] Figure 6 and 7 A flowchart illustrating a method for photomask measurement techniques supporting anti-flatness errors according to various aspects of this disclosure is shown. Detailed Implementation

[0016] Semiconductor materials can be patterned using optical lithography to facilitate manufacturing operations for processing the semiconductor material (e.g., during the formation of electronic devices). A substrate, such as a photomask (e.g., a photolithographic mask, photomask), can be implemented to transfer a pattern (e.g., pattern data, image) to the semiconductor material, exposing the semiconductor material to the pattern (e.g., marking). After the semiconductor material is patterned via the substrate, one or more manufacturing operations (e.g., etching, deposition) can be performed based on the pattern to process the semiconductor material and ultimately produce an electronic device. However, deviations in the flatness (e.g., planarity) of the substrate (e.g., deviations from an ideal plane, deviations from desired flatness) can introduce related deviations (e.g., errors) in the patterning of the semiconductor material during pattern transfer. Therefore, it may be desirable to measure the flatness of the substrate before patterning the semiconductor material so that deviations in flatness can be identified and accounted for (e.g., adjusted) during the patterning of the semiconductor material.

[0017] In some embodiments, determining the flatness of a substrate may include holding the substrate in close proximity to a reference surface having a known flatness. In some such embodiments, a laser source (e.g., a coherent laser source) may illuminate the reference surface and the substrate surface at least partially overlapping (at least partially simultaneous) in time, and an interferometer may measure the flatness of the substrate surface by comparing the substrate surface and the reference surface. In some examples, a controller may generate one or more interferograms for performing phase measurement interferometry (PMI) on the substrate.

[0018] In the embodiments, when measuring and determining the flatness of a photomask, the flatness deviation (error in the flatness measurement) caused by gravity acting on the photomask can be relatively reduced or eliminated. That is, the system disclosed herein can separate the flatness deviation of the photomask caused by gravity from the actual flatness of the photomask, where the actual flatness is the true flatness of the photomask. Therefore, in the embodiments disclosed herein, the gravity component of the flatness measurement is removed from the flatness measurement to provide the true flatness of the photomask.

[0019] In an embodiment, the flatness of the substrate can be measured in one or more orientations, and the measurements can then be fitted to an orthogonal polynomial (such as a Zernike polynomial) to generate an error estimate associated with gravity. The system can remove the error estimate from the flatness measurements to determine an error-resistant measurement of the substrate's flatness (true flatness). As is known in the art, Zernike polynomials can be used to mathematically describe 3-D wavefront deviations. Each Zernike polynomial describes a specific form of surface deviation, allowing the combination of two or more Zernike polynomials to describe more complex surface shapes. In principle, by including a sufficient number of Zernike polynomials (often referred to as "terms"), any wavefront deformation can be described with the desired level of accuracy. Each Zernike term is referenced by a polynomial order number (0, 1, 2, 3, 4, 5, 6, 7, etc.) and grouped according to angular frequency (θ) and radial order. Furthermore, as is known in the art, Zelnique polynomials can be referred to as “even” or “odd”, such that an even Zelnique polynomial corresponds to a polynomial with an even number of angular frequencies (0θ, 2θ, 4θ, 6θ, etc.), and an odd Zelnique polynomial corresponds to a polynomial with an odd number of angular frequencies (1θ, 3θ, 5θ, 7θ, etc.).

[0020] The aspects of this disclosure are initially described in the context of a system implementing a photomask measurement technique resistant to flatness errors. Further aspects of this disclosure are described in the context of photomask orientation diagrams and process flows for a system implementing a photomask measurement technique resistant to flatness errors. The aspects of this disclosure are further illustrated by block diagrams and flowcharts relating to photomask measurement techniques resistant to flatness errors, and are described with reference to said block diagrams and flowcharts.

[0021] This description provides examples and is not intended to limit the scope, applicability, or configuration of the principles described herein. Rather, the following description will provide those skilled in the art with a feasible description for implementing various aspects of the principles described herein. As will be appreciated by those skilled in the art, various changes can be made to the function and arrangement of the elements without departing from this application.

[0022] Those skilled in the art will understand that one or more aspects of this disclosure can be implemented in a system to additionally or alternatively solve problems other than those described herein. Furthermore, aspects of this disclosure can provide technical improvements to “conventional” systems or processes as described herein. However, the specification and drawings only include exemplary technical improvements resulting from implementing aspects of this disclosure and therefore do not represent all technical improvements provided within the scope of the claims and this disclosure.

[0023] Figure 1 An example of a system 100 providing error-resistant photomask measurements for different support locations of a photomask according to aspects of this disclosure is shown. Specifically, system 100 includes various measuring components operable to measure the flatness of a substrate 130. For illustrative purposes, aspects of system 100 may be described with reference to the x, y, and z directions of the illustrated coordinate system. For example, system 100 may be shown as a side view in the yz plane. In some examples, the y direction may indicate the direction opposite to gravity, and each measuring component may extend a distance along the x direction. Although system 100 shows examples of the relative dimensions and numbers of various features, aspects of system 100 may be implemented using other relative dimensions or numbers of such features as in the examples disclosed herein.

[0024] The substrate 130 can be, for example, a photomask or a blank surface. The substrate 130 can be a coated glass material or an uncoated glass material. Figure 1 As shown, substrate 130 includes a first surface 132, a second surface 134, a top surface 136, and a bottom surface 138. System 100 can measure the flatness of the first surface 132 and / or the second surface 134.

[0025] System 100 may include a controller 105 configured to facilitate operation of system 100. Controller 105 may include one or more processors configured to perform operations based on configurations, algorithms, or programs stored in controller 105 or an external memory system. For example, controller 105 may include a non-transitory computer-readable medium configured to store instructions including code operable to cause controller 105 to perform operations related to system 100. Controller 105 may be configured to communicate signaling with interferometer 145 via, for example, one or more interfaces. For example, controller 105 may communicate one or more commands to interferometer 145 to measure the flatness of substrate 130 (e.g., to collect measurement data associated with the shape of substrate 130). Additionally, controller 105 may receive measurements of the flatness of substrate 130 from interferometer 145. Interferometer 145 may be, for example, a Fizeau interferometer 145 configured to measure the flatness of substrate 130 via a number of components. For example, interferometer 145 may include components associated with generating and aiming light at substrate 130, and components associated with comparing light from substrate 130 with reference surface 125 to determine flatness deviations of substrate 130. In some embodiments, system 100 may include more than one interferometer 145.

[0026] Controller 105 can be configured to communicate with interferometer 145 via signaling from laser 110 (e.g., laser source, laser diode) through interface 106 (e.g., bus) coupling controller 105 to laser 110. For example, controller 105 can (via interface 106) transmit one or more command communications to laser 110 to initiate the execution of one or more operations associated with measuring the flatness of substrate 130. Laser 110 can be configured to output one or more laser beams 115 associated with measuring the flatness of substrate 130. Figure 1As shown, interferometer 145 may include a focusing element 111 configured to focus one or more laser beams 115 onto one or more optical elements 120. The one or more optical elements 120 of the interferometer may be configured to direct one or more laser beams 115 onto a reference surface 125 or a first surface 132 of the substrate 130, or both. In some cases, the one or more optical elements 120 may be configured to guide, refract, and / or focus one or more laser beams 115 such that the one or more laser beams 115 illuminate the entire surface (or a portion thereof) of the reference surface 125 and / or the first surface 132 of the substrate 130. For example, laser 110 may output a laser beam 115, which may be focused by the focusing element 111 to align the laser beam 115 with the optical element 120, which may project the laser beam 115 onto the reference surface 125 and the substrate 130.

[0027] Reference surface 125, also known as a Fizeau surface, is a relatively large flat surface with known flatness. As discussed further below, reference surface 125 serves as a reference for comparison with substrate 130 to determine the flatness of substrate 130. In some embodiments, reference surface 125 may comprise fused silica.

[0028] For example Figure 1 As shown, the interferometer 145 further includes a support 140 to hold the substrate 130 in a desired position with a desired orientation. The support 140 includes at least two support members 135-a, 135-b. Figure 2A-2D As shown, support member 135-a may comprise two support members 135-a-1 and 135a-2, each positioned along the bottom surface of substrate 130. Furthermore, support member 135-b may be positioned along the top surface of substrate 130. However, it is also contemplated that support members 135-a-1, 135-a-2, and 135-b may be positioned along different portions and / or surfaces of substrate 130, rather than as specifically disclosed herein. Figure 1 As shown, support members 135-a-1, 135-a-2, and 135-b are each cantilever members extending from the base 142 of the support member 140. In the embodiments disclosed herein, support members 135-a-1 and 135-a-2 directly contact only the bottom surface 138 and the chamfered edge at the bottom of the substrate 130. Furthermore, support member 135-b directly contacts only the chamfered edge at the top of the substrate 130. Therefore, support members 135-a-1, 135-a-2, and 135-b do not directly contact the first surface 132 or the second surface 134 of the substrate 130.

[0029] Support members 135-a and 135-b are configured to maintain a distance between the substrate 130 and the base 142 of the support member 140. In some embodiments, the second surface 134 of the substrate 30 is maintained at a distance from the base 142 of the support member 140, while being parallel to the base 142 of the support member 140.

[0030] The support 140 can be held at an angle 141 relative to the vertical direction of the y-axis. Therefore, the base 142 can also be held at an angle 141 relative to the vertical direction of the y-axis. Because the support 140 is held at this angle, the substrate 130 is also held at an angle 141 relative to the vertical direction of the y-axis. Specifically, angle 141 is defined herein as the angle between the second surface 134 of the substrate 130 and the vertical direction of the y-axis. Furthermore, angle 141 is also referred to herein as the “tilt angle” of the substrate 130. The value of angle 141 can be adjusted to change the magnitude of gravity acting on the substrate 130, such that a larger value of angle 141 corresponds to a greater tilt of the substrate 130 relative to the y-axis, while a smaller value of angle 141 corresponds to a more vertical orientation of the substrate 130 along the y-axis.

[0031] When substrate 130 is in a horizontal position, gravity acts on substrate 130 in a direction orthogonal to surface 132 of substrate 130. When substrate 130 is more vertically oriented (when the value of angle 141 is relatively small), the magnitude of gravity acting on substrate 130 is relatively small. Furthermore, the flatness measurement of substrate 130 may be affected by the magnitude of gravity acting on substrate 130. Substrate 130 is not a perfectly rigid component, and therefore, when in a horizontal position, it is prone to deviation due to gravity. In other words, when substrate 130 is in a horizontal position, the first surface 132 and the second surface 134 of substrate 130 may be slightly concave due to gravity. Although the deviation may be very small, this deviation may cause conventional interferometers to inaccurately measure the flatness of the substrate. Conventional interferometers may inaccurately interpret this deviation caused by gravity as part of the true flatness of the substrate. However, when the substrate 130 is oriented in a more vertical position (when the value of angle 141 is relatively small), it is less susceptible to the effects of gravity and therefore does not have this concavity (or has a reduced amount of concavity). Because the magnitude of gravity acting on the substrate 130 is smaller when the substrate is more vertically oriented, the substrate is also less prone to errors in flatness measurements caused by gravity.

[0032] While a more vertical orientation of the substrate 130 is beneficial because it provides more accurate flatness measurements, this must be balanced with the stability of the substrate 130. As the value of angle 141 decreases, the substrate 130 may experience relatively more vibration and greater instability, making it more prone to falling off the support 140. Therefore, the value of angle 141 should be balanced to provide accurate measurements while preventing the substrate 130 from falling off the support 140. In an embodiment, the value of angle 141 may be maintained between about 1 degree and about 5 degrees (to provide a desired balance between stability and the effects of gravity). Support members 135-a-1, 135-a-2, and 135-b, together with the base 142, hold the substrate 130 at a desired angle 141 between about 1 degree and about 5 degrees. It is worth noting that even within these ranges of the desired angle 141, the substrate 130 may still experience some deviation due to gravity, even if small, for example, a deviation of about 20 nm to about 50 nm.

[0033] In an embodiment, the interferometer 145 may include one or more light-receiving components such that light reflected from the first surface 132 of the substrate 130 can be compared with light reflected from the reference surface 125 to determine the shape difference between the first surface 132 and the reference surface 125 of the substrate 130 (in order to ultimately measure the true flatness of the substrate 130). For example, the interferometer 145 may include a detector array that may include one or more cameras configured to receive light directed from the first surface 132 or the reference surface 125 of the substrate 130, or both. In some such examples, the interferometer 145 may include one or more refractory lenses configured to direct light reflected from the first surface 132 or the reference surface 125 of the substrate 130, or both, to the detector array.

[0034] Controller 105 is configured to modify one or more parameters of interferometer 145 to determine the shape and flatness of the first surface 132 of substrate 130. For example, controller 105 may modify the brightness, wavelength, spatial coherence, or illumination angle of laser 110. Alternatively or additionally, controller 105 may modify the number of measurement frames, the size of each measurement frame, or the integration time of the detector array. Alternatively or additionally, controller 105 may control the movement of support 140 to modify the position or alignment of support 140 (and therefore, also modify the position or alignment of substrate 130 relative to reference surface 125). For example, controller 105 may pivot support 140 to change angle 141 of substrate 130. In some embodiments, controller 105 may activate other subsystems (such as a robot) to remove and reposition substrate 130 to rotate substrate 130 in its plane, for example, by 90-degree increments.

[0035] The flatness measurement of the first surface 132 of substrate 130 includes an assessment of the flatness of the first surface 132 and any deformation of this flatness relative to a perfectly flat ideal plane. Substrate 130 itself contains its true flatness, which is merely a measurement of the flatness of the surface of substrate 130. However, the gravity acting on the substrate together with the support points 135-a and 135-b can cause a deviation from the true flatness of the substrate. This force can cause the flatness measurement to deviate from its true flatness. More specifically, the magnitude of the gravity acting on the substrate may cause a slight indentation of the substrate, resulting in a larger deviation of the flatness measurement from an ideal plane. Therefore, in this case, the measured flatness of the substrate may differ from the true flatness of the substrate. As discussed further below, measurement errors in the interferometer itself can also cause deviations (errors) in the true flatness measurement of the substrate. As discussed above, accurate measurement of the flatness of the substrate is important for patterning semiconductor materials using the substrate. Therefore, deviations in flatness measurements (such as those caused by gravity acting on the substrate or measurement errors in the interferometer) can introduce unwanted deviations when patterning semiconductor materials.

[0036] According to the embodiments disclosed herein, system 100 is configured to determine the true flatness of substrate 130 while relatively reducing or eliminating flatness deviations (errors in flatness measurements) caused by gravity acting on substrate 130 or measurement errors from the interferometer. In other words, system 100 can separate the flatness deviation of substrate 130 caused by gravity from the true flatness of substrate 130. Therefore, in the embodiments disclosed herein, the gravity component and the interferometer measurement error component of the flatness measurement are removed from the flatness measurement value to provide the true flatness of the substrate.

[0037] According to the embodiments disclosed herein, interferometer 145 measures the flatness of substrate 130 when it is in a non-horizontal position to prevent (or reduce) any deviation of the substrate caused by gravity. Furthermore, as further discussed below, interferometer 145 measures the flatness of substrate 130 in one or more orientations of the substrate and at a tilt angle 141. After measuring the flatness of substrate 130 in one or more orientations, system 100 (e.g., controller 105) extracts, for example, a Zernike factor from the measurements and uses the Zernike factor to generate an error estimate associated with gravity. The system can remove the error estimate from the flatness measurements to determine an error-resistant measurement of the flatness (true flatness) of substrate 130. Therefore, compared to conventional interferometers, system 100 can provide a more accurate measurement of the true flatness of substrate 130, which can then be used to modify the positioning of patterned data on substrate 130, allowing for more accurate transfer of patterned data to the semiconductor material during patterning. In addition, system 100 can be configured to take into account the rotational symmetry error of the flatness measurement of substrate 130, which conventional interferometers may not be able to account for in other respects.

[0038] Figure 2A-2D An orientation diagram illustrating an embodiment of this disclosure is shown. Error! Examples of reference sources 00-a, 200-b, 200-c, and 200-d not found. See reference... Figure 1 As described, the orientation chart can be implemented by or by various aspects of system 100. The orientation chart depicts various orientations for measuring the flatness of substrate 130. For illustrative purposes, aspects of the orientation chart can be described with reference to the x, y, and z directions of the illustrated coordinate system. For example, orientation charts 200-a, 200-b, 200-c, and 200-d can show a front view of substrate 130 in the xy plane, showing the first surface 132 of substrate 130. In some examples, the y direction can indicate the direction opposite to gravity, and each of the components can extend a distance along either the x or z direction. Although the orientation chart shows examples of the relative dimensions and quantities of various features, aspects of the orientation chart can be implemented using other relative dimensions or quantities of such features, as illustrated in the examples disclosed herein.

[0039] The flatness of the surface of substrate 130 can be measured at a certain tilt angle in one or more orientations. Each of the different orientations of substrate 130 can be separated by a corresponding degree angle (e.g., 90 degrees), such that each orientation can represent a 90-degree increment of rotation along an axis perpendicular to the surface of substrate 130 (e.g., an axis in the z-direction). In other embodiments, the corresponding degree angles for the different orientations are, for example, 45 degrees, 180 degrees, and 270 degrees. However, angles other than those specifically disclosed herein can be used. A robotic mechanism (not shown) can disengage substrate 130 from support 140 to rotate the substrate to a different orientation and then reposition substrate 130 onto support 140. Figure 2A As shown, substrate 130 is in a first orientation. In this example, the robotic mechanism then picks up substrate 130 and removes substrate 130 from support 140. Then, as... Figure 2B As shown, the robot mechanism rotates the substrate 130 90 degrees counterclockwise and repositions the substrate 130 onto the support 140. Therefore, Figure 2B The substrate 130 is shown in a second orientation after rotation. A robotic mechanism can also be used to rotate the substrate 130 to... Figure 2C The third orientation and Figure 2D The fourth orientation. For illustrative purposes, Figure 2A-2D Reference points are shown to depict the rotation of substrate 130. As further discussed below, interferometer 145 measures the flatness of substrate 130 at each orientation. In some embodiments, interferometer 145 measures substrate 130 at four different orientations.

[0040] In addition, such as Figure 2A-2D Each of the first, second, third, and fourth orientations shown can be at the same tilt angle 141. Therefore, the substrate 130 can maintain the same angle in each of the orientations. In an embodiment, the tilt angle 141 is about 1 degree to about 5 degrees, or about 2 degrees to about 4 degrees, or about 3 degrees to about 5 degrees.

[0041] The flatness of substrate 130 can be measured under different orientations via interferometer 145 to generate corresponding flatness measurement values ​​for substrate 130. For example, the flatness of substrate 130 can be measured under a first orientation to generate a first flatness measurement value for substrate 130, under a second orientation to generate a second flatness measurement value, under a third orientation to generate a third flatness measurement value, and under a fourth orientation to generate a fourth flatness measurement value. Controller 105 can use one or more of the first, second, third, and / or fourth flatness measurement values ​​to generate the true flatness of substrate 130, wherein flatness deviations of substrate 130 caused by gravity are removed. As discussed above, each of the first, second, third, and fourth orientations can be rotated 90 degrees relative to each other. Therefore, in the embodiments, the first orientation is 0 degrees, the second orientation is 90 degrees, the third orientation is 180 degrees, and the fourth orientation is 270 degrees. Figure 2A-2D As shown, these orientations can be referenced to reference points.

[0042] Each of the first, second, third, and fourth flatness measurements may include a deviation (error), which is removed when calculating the final true flatness of the substrate 130. For example, the substrate 130 will be subjected to gravity when mounted on the support 140 (even if the substrate is not in a horizontal position, it may still experience some minor deviations (dips) caused by gravity). Gravity causes a deviation in each of the first, second, third, and fourth flatness measurements. Furthermore, the interferometer 145 may have internal inaccuracies (even small ones), which may contribute to the measured flatness of the substrate. As an example, wavefront errors in the interferometer 145 may affect the measured flatness. In the embodiments disclosed herein, deviations and inaccuracies in the first, second, third, and fourth flatness measurements are identified and removed to provide the true flatness of the substrate 130.

[0043] In equation (1) below, M represents the true flatness of substrate 130 without the influence of gravity or any other wavefront error, G represents the deviation of the true flatness caused by gravity, and W represents the deviation of the true flatness caused by interferometer 145 (such as wavefront error from the interferometer). The sum of M, G, and W is the measured flatness F. In an embodiment, F is the flatness measured by interferometer 145. As discussed further below, in embodiments of this disclosure, the deviation of the true flatness caused by gravity (G) is determined and said deviation is removed from the measured flatness (F) to determine the true flatness (M) of the substrate.

[0044] F = M + G + W (1)

[0045] It is worth noting that, in the embodiment, the bias error caused by the interferometer (W) can be removed during the calibration process of the interferometer 145, so the value of W may be relatively small (e.g., about 3 nm or less).

[0046] Figure 3A The process for measuring the true flatness (M) of a substrate 130 according to an embodiment of this disclosure is shown in the image. Error! Reference source not found. 00. Process 300 can demonstrate various aspects of measuring the true flatness of the substrate 130, as shown in the reference... Figure 1-2D As described herein, the steps of process 300 may be performed in a different order or at different times. For example, some of the steps and techniques described herein may be rearranged, arranged, or otherwise rearranged to produce similar results, and the examples provided herein should not be considered as limiting the scope of this disclosure or the claims. For example, and as described below, one or more aspects of this disclosure may be performed via a different order or combination of steps than that described herein.

[0047] At step 305 of process 300, interferometer 145 measures the flatness of substrate 130 under one or more different orientations to generate four flatness measurements. In an embodiment, interferometer 145 measures the flatness of substrate 130 under each of a first orientation, a second orientation, a third orientation, and a fourth orientation to generate a first flatness measurement, a second flatness measurement, a third flatness measurement, and a fourth flatness measurement, respectively. The flatness measurements generated in step 305 can be referenced herein. Figure 1 The measurement is performed so that system 100 can perform the measurement. For example, interferometer 145 can measure and compare light detected from one or more laser beams 115 illuminating the surface of reference surface 125 and the first surface 132 of substrate 130.

[0048] It is worth noting that the first, second, third, and fourth flatness measurements each correspond to the F term in equation (1) above. Specifically, as shown in equations (2) to (5) below, F1 corresponds to the first flatness measurement under the first orientation (0 degrees), F2 corresponds to the second flatness measurement under the second orientation (90 degrees), F3 corresponds to the third flatness measurement under the third orientation (180 degrees), and F4 corresponds to the fourth flatness measurement under the fourth orientation (270 degrees). Furthermore, G in each of equations (2) to (5) corresponds to the deviation of the actual flatness caused by gravity.

[0049] F1 = M0 + G + W (2)

[0050] F2 = M 90 + G + W (3)

[0051] F3 = M 180 + G + W (4)

[0052] F4 = M 270 + G + W (5)

[0053] In the above equation, process 300 is used to determine the G and W terms and remove the G and W terms from the F term to determine the true flatness of the substrate (M term). To determine the G term, different flatness measurements generated in this paper under different orientations are mapped and compared with each other to determine the effect of gravity on the substrate.

[0054] At step 310 of process 300, the controller can generate a first set of differences by subtracting one of the following flatness measurements from one of the first, second, third, and fourth flatness measurements. However, the flatness measurements subtracted in step 310 should be rotated 90 degrees relative to each other. Therefore, in an embodiment of step 310, the first flatness measurement (corresponding to a 0-degree orientation) is subtracted from the second flatness measurement (corresponding to a 90-degree orientation) because the orientations corresponding to these flatness measurements are rotated 90 degrees relative to each other. In another embodiment of step 310, the second flatness measurement (corresponding to a 90-degree orientation) is subtracted from the third flatness measurement (corresponding to a 180-degree orientation) because the orientations corresponding to these flatness measurements are rotated 90 degrees relative to each other. In an embodiment of step 310, the third flatness measurement value (corresponding to a 180-degree orientation) is subtracted from the fourth flatness measurement value (corresponding to a 270-degree orientation) because the orientations corresponding to these flatness measurements are rotated 90 degrees relative to each other. In another embodiment of step 310, the fourth flatness measurement value (corresponding to a 270-degree orientation) is subtracted from the first flatness measurement value (corresponding to a 0-degree orientation) because the orientations corresponding to these flatness measurements are rotated 90 degrees relative to each other.

[0055] At step 315 of process 300, the controller can generate a second set of differences by subtracting one of the following flatness measurements from the first, second, third, and fourth flatness measurements. However, the flatness measurements subtracted in step 315 should be rotated 180 degrees relative to each other. Therefore, in the embodiment of step 310, the first flatness measurement (corresponding to the 0-degree orientation) is subtracted from the third flatness measurement (corresponding to the 180-degree orientation) because the orientations corresponding to these flatness measurements are rotated 180 degrees relative to each other. In the embodiment of step 310, the second flatness measurement (corresponding to the 90-degree orientation) is subtracted from the fourth flatness measurement (corresponding to the 270-degree orientation) because the orientations corresponding to these flatness measurements are rotated 180 degrees relative to each other.

[0056] It is worth noting that in each of steps 310 and 315, the controller 105 may perform one or more subtraction calculations between different flatness measurements. For example, in step 310, the controller 105 may subtract the first flatness measurement value from only the second flatness measurement value to generate a first set of differences. In other embodiments of step 310, the controller 105 may subtract the first flatness measurement value from the second flatness measurement value and subtract the second flatness measurement value from the third flatness measurement value to generate a first set of differences. In yet another embodiment of step 310, the controller 105 may subtract the first flatness measurement value from the second flatness measurement value, subtract the second flatness measurement value from the third flatness measurement value, subtract the third flatness measurement value from the fourth flatness measurement value, and subtract the fourth flatness measurement value from the first flatness measurement value to generate a first set of differences. It is worth noting that the more subtraction calculations performed in each of steps 310 and 315, the more data is generated, thus enabling a more accurate determination of the true flatness of the substrate 130.

[0057] The various steps of process 300 (such as steps 320 and 325) use orthogonal polynomials to characterize and describe the measurements calculated herein. In embodiments, the orthogonal polynomials utilized may include, for example, Zelnik polynomials, Legendre polynomials, and XY polynomials. For the purposes of the steps in process 300, such as Figure 3A The Zernike polynomial is shown and described in the diagram. However, the steps of procedure 300 should not be limited to this Zernike polynomial, and other orthogonal polynomials can be used.

[0058] At step 320 of process 300, the first set of differences (as calculated in step 310 above) is fitted to an Ozeirnik polynomial. More specifically, the differences calculated in each subtraction calculation in step 310 are fitted to an Ozeirnik polynomial. In one example, the first set of differences in step 310 includes the following subtraction calculations: (i) subtracting the first smoothness measurement from the second smoothness measurement, (ii) subtracting the second smoothness measurement from the third smoothness measurement, (iii) subtracting the third smoothness measurement from the fourth smoothness measurement, and (iv) subtracting the fourth smoothness measurement from the first smoothness measurement. The calculations of (i)-(iv) are further shown in the calculations (6)-(9) below:

[0059] F 90 – F0 = (M 90 + G + W) – (M0 + G1 + W) = M 90 – M0(6)

[0060] F180 – F 90 = (M 180 + G + W) – (M 90 + G + W) = M 180 – M 90 (7)

[0061] F 270 – F 180 = (M 270 + G + W) – (M 180 + G + W) = M 270 – M 180 (8)

[0062] F0 – F 270 = (M0 + G + W) – (M 270 + G + W) = M0 – M 270 (9)

[0063] Then, in step 320, M 90 – M0、M 180 – M 90 M 270 – M 180 and M0 – M 270 Each mapping is fitted to an even-Zernik polynomial. Specifically, in the embodiment, M is... 90 – M0、M 180 – M 90 M 270 – M 180 and M0 – M 270 The mappings are each fitted to 2θ and 6θ Zernike polynomials to generate even Zernike polynomials for each mapping. In these embodiments, even angular frequencies above 6θ have very little effect on the shape of the polynomials and are therefore not used. Furthermore, in step 320, even if these polynomials are "even" polynomials, the mappings are not fitted to 0θ and 4θ Zernike polynomials because the 0θ and 4θ Zernike polynomials are 90-degree symmetric and therefore would cancel each other out in the subtraction described above.

[0064] Figure 3B An example M is shown. 90 – The M0 mapping, which is fitted to 2θ and 6θ Zernike polynomials to generate even Zernike polynomials for this mapping. Specifically, Figure 3B M0 and M are shown in the top row of this figure. 90 Mapping and showing M in the second line 90– M0 mapping. The third row of this figure shows the M0 mapping fitted to the 2θ Zernike polynomial. 90 – M0 mapping and fitting to the 6θ Zernick polynomial M 90 – M0 mapping. It's worth noting that M... 90 – The M0 mapping looks neither like M 90 The mapping is also unlike the M0 mapping because it does not have any errors associated with the deviation of the true flatness caused by gravity G or the deviation of the true flatness caused by the interferometer W, as indicated in calculations 6-9 above. It is also worth noting that the error from M... 90 – The 2θ and 6θ terms fitted to the M0 mapping are the same as those of the M0 mapping (but with different scaling factors), as is known to those skilled in the art.

[0065] Then, in step 320, the even-Zernik polynomials generated for each mapping are averaged together. In some exemplary embodiments, in step 320, all calculations (6), (7), (8), and (9) can be used to generate a first set of differences, which are used to generate even-Zernik polynomials. However, as mentioned above, only some of the calculations in (6), (7), (8), and (9) can be used to generate the first set of differences.

[0066] At step 325 of process 300, the second set of differences (as calculated in step 315 above) is fitted to a Kieselnik polynomial. More specifically, the differences calculated in each subtraction calculation in step 315 are fitted to a Kieselnik polynomial. In one example, the second set of differences in step 310 includes the following subtraction calculations: (v) subtracting the first smoothness measurement from the third smoothness measurement, and (vi) subtracting the second smoothness measurement from the fourth smoothness measurement. The calculations of (v) and (vi) are further shown in calculations (10) and (11) below:

[0067] F 180 – F0 = (M 180 (+ G + W) – (M0 + G + W) = M 180 – M0 (10)

[0068] F 270 – F 90 = (M 270 + G + W) – (M 90 + G + W) = M 270 – M 90 (11)

[0069] Then, in step 320, M 180– M0 and M 270 – M 90 Each mapping is fitted to a Chisernik polynomial. Specifically, M... 180 – M0 and M 270 – M 90 Each mapping is fitted to a 1θ, 3θ, and 5θ Zernike polynomial to generate a chizelnik polynomial for each mapping. In these embodiments, even angular frequencies above 5θ have very little effect on the shape of the polynomial and are therefore not used.

[0070] Figure 3C An example M is shown. 180 – The M0 mapping is fitted to 1θ, 3θ, and 5θ Zernike polynomials to generate chizelk polynomials for this mapping. Specifically, Figure 3C M0 and M are shown in the top row of this figure. 180 Mapping and showing M in the second line 180 – M0 mapping. The third row of this figure shows the M0 mapping fitted to the 1θ, 3θ, and 5θ Zernike polynomials. 180 – M0 mapping. It's worth noting that M... 180 – The M0 mapping looks neither like M 180 The mapping is also unlike the M0 mapping because it does not have any errors associated with deviations in true flatness caused by gravity G or by the interferometer W, as indicated in calculations 10 and 11 above. It is also worth noting that the error from M... 180 – The 1θ, 3θ, and 5θ terms fitted to the M0 mapping are the same as those of the M0 mapping (but with different scaling factors), as is known to those skilled in the art.

[0071] Then, the Chizernik polynomials generated for each mapping are averaged together. In some exemplary embodiments, in step 325, both calculations (10) and (11) can be used to generate a second set of differences, which are used to generate the Chizernik polynomials. However, as mentioned above, only one of calculations (10) and (11) can be used to generate the second set of differences.

[0072] At step 330, controller 105 can combine the average even-Zernik polynomial from step 320 with the average chi-Zernik polynomial from step 325 by averaging the polynomials together to generate a fifth flatness measurement. The fifth flatness measurement may include angular frequencies of 1θ, 2θ, 3θ, 5θ, and 6θ (note that the 2θ and 6θ angular frequencies are from step 320, and the 1θ, 3θ, and 5θ angular frequencies are from step 325). In some embodiments, controller 105 may rotate the Zernik polynomials to align them before combining the average even-Zernik polynomial from step 325 with the average chi-Zernik polynomial from step 325.

[0073] At step 335, one of the following flatness measurements—a first flatness measurement, a second flatness measurement, a third flatness measurement, and a fourth flatness measurement—is selected, and then further processed in subsequent steps of process 300. In some embodiments, the controller 105 selects the flatness measurement in step 335, while in other embodiments, the user selects the flatness measurement in step 335. The controller 105 may randomly select the flatness measurement in step 335, or it may select it based on one or more factors. In yet another embodiment, more than one flatness measurement is selected in step 335, and the flatness measurements are averaged together. In an exemplary embodiment, a first flatness measurement F1 is selected in step 335.

[0074] At step 340, controller 105 can remove even and odd Zernik polynomials from the selected flatness measurement value in step 335, except for the 0θ and 4θ Zernik terms. Therefore, the 0θ and 4θ Zernik terms remain after step 340 is completed. In an exemplary example, controller 105 selects a first flatness measurement value F1 in step 335, and in step 340, removes each of the 1θ, 2θ, 3θ, 5θ, and 6θ Zernik terms from the first flatness measurement value F1. Therefore, the 0θ and 4θ Zernik terms remain when step 340 is completed with respect to the first flatness measurement value F1.

[0075] It is worth noting that the F1 mapping does indeed contain errors associated with the deviations in true flatness caused by gravity (G) and the interferometer (W), as indicated in the calculation (2) above. It is also worth noting that the fitted 1θ, 2θ, 3θ, 5θ, and 6θ terms contain errors associated with the deviations in true flatness caused by gravity (G) and the interferometer (W), which are removed from the first flatness measurement F1 when the Zernike terms are removed in step 340.

[0076] At step 345, controller 105 replaces the fitted 1θ, 2θ, 3θ, 5θ, and 6θ terms from step 340 containing errors associated with G and W with the fitted 1θ, 2θ, 3θ, 5θ, and 6θ terms from step 330 (the fifth flatness measurement) that do not contain errors associated with G and W, to generate the sixth flatness measurement. As mentioned above, the fifth flatness measurement includes 1θ, 2θ, 3θ, 5θ, and 6θ Zernike terms. Therefore, by replacing the Zernike terms of the flatness measurement selected in step 335 with the corresponding fifth flatness measurement terms, the generated sixth flatness measurement includes 1θ, 2θ, 3θ, 4θ, 5θ, and 6θ Zernike terms, from which the effects of gravity have been removed.

[0077] The sixth flatness measurement in step 345 is the estimated true flatness M of the substrate 130 corresponding to the orientation of the flatness selected in step 335. However, this estimated true flatness is further improved and calculated more accurately through subsequent steps of process 300. In an embodiment, when the first flatness measurement F1 is selected in step 335, the sixth flatness measurement is the estimated true flatness of M0 (wherein the estimated true flatness of this orientation is referred to below as M). 0' In this embodiment, when the second flatness measurement value F2 is selected in step 335, the sixth flatness measurement value is M. 90 The estimated true flatness. In the embodiment, when the third flatness measurement value F3 is selected in step 335, the sixth flatness measurement value is M. 180 The estimated true flatness. In the embodiment, when the fourth flatness measurement value F4 is selected in step 335, the sixth flatness measurement value is M. 270 The estimated true flatness.

[0078] To further improve and more accurately calculate the estimated true flatness in step 350, in steps 350-360 of process 300, the deviation in true flatness caused by gravity G is removed from the estimate. At step 350, controller 105 subtracts the flatness measurement value selected in step 335 from the sixth flatness measurement value to generate a seventh flatness measurement value for the substrate. As described above, in this embodiment, a first flatness measurement value F1 is selected in step 335 such that the sixth flatness is the estimated true flatness M. 0' Therefore, in these embodiments, the sixth flatness measurement value M is used. 0' The first flatness measurement value F1 is subtracted from the first flatness measurement value to generate the seventh flatness measurement value. This calculation is shown in the calculation (12) below, such that the first flatness measurement value F1 contains M0 + G + W, as discussed above.

[0079] F1 – M 0' = (M0 + G + W) – M 0' = G(12)

[0080] In the calculation (12) above, M 0' This is an estimate of M0, which makes these values ​​considered to cancel each other out. Furthermore, the interferometer 145 can be calibrated so that W in the above calculation 12 becomes negligible. As shown in the above calculation (12), the sixth flatness measurement (M0) is subtracted from the first flatness measurement (F1). 0' The seventh flatness measurement is generated, which is equal to G (the deviation of the true flatness caused by gravity). Therefore, the seventh flatness measurement is equal to G, but this G term does not include the 0θ Zernike term. It is worth noting that the 0θ Zernike terms cancel each other out in the subtraction of (12) above, so that the resulting G is missing the 0θ Zernike term.

[0081] To correct for the 0θ Zernike term in the seventh flatness measurement, the inventors of this disclosure explored the relationship between a 5th-order Zernike polynomial (associated with the 2θ term) and a 4th-order Zernike polynomial (associated with the 0θ term). The inventors found that the 4th-order Zernike polynomial is correlated with the 5th-order Zernike polynomial, allowing the prediction of the 4th-order Zernike term from the 5th-order Zernike term. In support members 135-a and 135-b, as... Figure 2A-2D In the embodiment shown in the diagram, the 5th-order Zernik polynomial is related to the 4th-order Zernik polynomial as follows: 4th-order Zernik polynomial = 5th-order Zernik polynomial multiplied by -0.225 (Z4 = Z5 x -0.225). However, it is worth noting that this relationship may vary depending on the position of the support members 135a-135b.

[0082] At step 355, a 5th-order Zernike polynomial (also known as the Zernike astigmatism factor) is extracted from the seventh flatness measurement (which is equal to G, as discussed above) to determine a 4th-order Zernike polynomial (also known as the Zernike power factor) using the relations disclosed above. Therefore, by determining the 5th-order Zernike polynomial (associated with the 2θ term), the 4th-order Zernike polynomial (associated with the 0θ term) is also determined. At step 360 of process 300, the sixth flatness measurement (e.g., the estimated true flatness M) is used... 0' Subtracting the fourth-order Zernike polynomial (the Zernike defocus factor, which is associated with the 0θ term) from the original equation yields the true flatness M of substrate 130. The true flatness M generated in step 360 represents the true flatness of the substrate under conditions unaffected by gravity.

[0083] After generating the true flatness of substrate 130, the true flatness can be used to aid in patterning semiconductor materials using substrate 130. For example, after determining a deviation in the flatness of substrate 130, this deviation can be taken into account when patterning the semiconductor material. In some examples, taking into account the deviation may include modifying the pattern (e.g., pattern data) printed on substrate 130 to compensate for printing errors that may be caused by the deviation. In some examples, taking into account the deviation may include modifying the placement of patterning data on substrate 130 to reduce the impact of the flatness deviation when patterning the semiconductor material. After taking into account the deviation, the modified substrate 130 can be used to pattern the semiconductor material. Similarly, the flatness deviation can be transmitted to one or more processors for use in the substrate manufacturing process, so that the deviation can be reduced during manufacturing. For example, the surface of substrate 130 can be deterministically polished to remove the flatness deviation.

[0084] Figure 4 An example of a process 400 supporting a substrate measurement technique for resisting flatness errors according to various aspects of this disclosure is shown. Process flow 400 can demonstrate various aspects of measuring the flatness of substrate 130, as referenced... Figure 1 and 2A As described in 2D. The flatness of substrate 130 can be measured at various orientations as shown in the reference substrate orientation diagram, and various operations can be performed on the measured values ​​to determine the anti-gravity measurement value of the flatness of substrate 130. For example, process 400 demonstrates operations for distinguishing the effect of gravity on the flatness of substrate 130 from the actual flatness of substrate 130, and then using this distinction to pattern semiconductor material based on modifications to the substrate to account for the effect of gravity. In the following description of process 400, some methods, techniques, processes, and operations may be performed in a different order or at different times. For example, some steps in the steps described herein may be reordered or otherwise rearranged to produce similar results. Therefore, Figure 4 The process 400 shown and described herein can be a non-limiting example of a process for determining a robust measurement of the flatness of the substrate 130. Therefore, other embodiments of the steps described herein for generating similar results can be understood as being represented by the examples shown and described herein in process 400. Additionally, some operational steps may be omitted from process flow 400, or other operational steps may be added to process 400.

[0085] At 405, interferometer 145 measures the flatness of substrate 130 under two orientations to generate two of the following flatness measurements: a first flatness measurement, a second flatness measurement, a third flatness measurement, and a fourth flatness measurement, as shown in the reference. Figure 2A-2DThe substrate orientation diagram is shown. In some embodiments, the interferometer 145 measures the flatness of the substrate 130 under a first orientation to generate a first flatness measurement value F1, and measures the flatness of the substrate under a second orientation to generate a second flatness measurement value F2, such that the second orientation is rotated 90 degrees relative to the first orientation.

[0086] At step 410 of process 400, controller 105 can generate a difference measurement value by subtracting another of the first, second, third, and fourth flatness measurement values ​​measured in step 405 from one of the flatness measurement values ​​measured in step 405. In the above embodiment, the first flatness measurement value F1 and the second flatness measurement value F2 are measured in step 405. Therefore, in step 410, the first flatness measurement value F1 is subtracted from the second flatness measurement value F2. This calculation is shown by calculation (13) as follows:

[0087] F2 – F1 = (M 90 + G + W) – (M0 + G1 + W) = M 90 – M0 (13)

[0088] The various steps of process 400 (such as steps 415 and 425) use orthogonal polynomials to characterize and describe the measurements calculated herein. In embodiments, the orthogonal polynomials utilized may include, for example, Zelnik polynomials, Legendre polynomials, and XY polynomials. For the purposes of the steps in process 400, such as Figure 4 The Zernike polynomial is shown and described in the diagram. However, the steps of procedure 400 should not be limited to this Zernike polynomial, and other orthogonal polynomials can be used.

[0089] At step 415 of process 400, the difference measurement (as calculated in step 410) is fitted to a first even-Zernik polynomial, which in an embodiment is a fifth-order Zernik polynomial associated with the 2θ term. More specifically, in the above embodiment, M is... 90 – The M0 map fits to a fifth-order polynomial associated with the 2θ term.

[0090] It is worth noting that, although M 90 – The M0 mapping looks neither like M 90The mapping is also unlike the M0 mapping, but it does not have any errors associated with deviations in true flatness caused by gravity G or by the interferometer W, as indicated in calculation 13 above. It is also worth noting that the error from M... 90 – The fitted 2θ term of the M0 mapping is the same as that of the M0 mapping (but with a different scaling factor), as is known to those skilled in the art.

[0091] At step 420, one of the following flatness measurements—a first flatness measurement, a second flatness measurement, a third flatness measurement, and a fourth flatness measurement—is selected, and then further processed in subsequent steps of process 400. In some embodiments, the controller 105 selects the flatness measurement in step 420, while in other embodiments, the user selects the flatness measurement in step 420. The controller 105 may randomly select the flatness measurement in step 420, or it may select it based on one or more factors. In yet another embodiment, more than one flatness measurement is selected in step 420, and the flatness measurements are averaged together. In an exemplary embodiment, a first flatness measurement F1 is selected in step 420.

[0092] At step 425 of process 400, the smoothness measurement selected in step 420 is fitted to a second even-Zernik polynomial, which in this embodiment is a fifth-order Zernik polynomial associated with the 2θ term. More specifically, in the above embodiment, the first smoothness measurement F1 is fitted to a fifth-order Zernik polynomial associated with the 2θ term. It is noteworthy that the F1 mapping does indeed contain errors associated with the deviations in true smoothness caused by gravity G and the deviations in true smoothness caused by the interferometer W, as indicated in the calculation (2) above. It is also noteworthy that the fifth-order polynomial associated with the 2θ term also contains errors associated with the deviations in true smoothness caused by gravity (G) and the deviations in true smoothness caused by the interferometer (W).

[0093] At step 430, the first Zernike polynomial (fifth order polynomial) of step 415 is subtracted from the second Zernike polynomial (fifth order polynomial) of step 425 to generate a fifth Zernike polynomial associated with the effect of gravity G on the flatness of the substrate.

[0094] At step 435, controller 105 can generate an error estimate by scaling the gravity error prediction model. The gravity error prediction model is generated using a finite element analysis (FEA) program (such as Ansys simulation software) to be based on... Figure 2A-2DThe simulation of the gravity acting on the substrate by the support members 135a, 135 shown is used to predict gravity-induced errors. The scaling factor is the ratio of the fifth-order Zernike polynomial from step 430 to the fifth-order polynomial fitted to the gravity error prediction model. This scaling factor can then be applied to the prediction model so that it correctly matches the expected error caused by the effects of gravity.

[0095] At step 440, the controller 105 can subtract the error estimate from step 435 from the flatness measurement value selected in step 420 to generate the true flatness M of the substrate. In an embodiment, a first flatness measurement value F1 is selected in step 420 such that the error estimate is subtracted from the first flatness measurement value F1 in step 440. This provides the true flatness M of the substrate 130 under conditions unaffected by gravity.

[0096] In an embodiment, the controller 105 can repeat the steps of process 400 to apply these steps to more than one selected flatness measurement. In one example, the controller 105 can apply the steps of process 400 in a first application of process 400, where a first flatness measurement F1 is selected in step 420, and can apply the steps of process 400 in a second application of process 400, where a second flatness measurement F2 is selected in step 420. The actual flatness measurements generated for each of the first and second applications in step 440 can then be averaged together. This can produce a more accurate determination of the actual flatness of the substrate 130.

[0097] Figure 5 A block diagram 500 shows a substrate measurement system 520 supporting error-resistant substrate measurement techniques for different support locations according to various aspects of this disclosure. The substrate measurement system 520 can be a reference... Figures 1 to 4 Examples of various aspects of the described substrate measurement system. The substrate measurement system 520 or its various components may be examples of apparatus for performing various aspects of error-resistant substrate measurement techniques for different support locations described herein. For example, the substrate measurement system 520 may include a positioning component 525, a measuring component 530, a generating component 535, and a patterning component 540, or any combination thereof. Each of these components may communicate with each other directly or indirectly (e.g., via one or more buses).

[0098] Positioning component 525 may be configured or otherwise supported for means of positioning the substrate in contact with one or more support points. Measurement component 530 may be configured or otherwise supported for means of measuring a first flatness measurement, a second flatness measurement, a third flatness measurement, and a fourth flatness measurement via one or more interferometers. Generation component 535 may be configured or otherwise supported for means of generating (in steps 310, 315, and / or 410) difference measurements and for fitting each difference measurement to a corresponding Zelnik polynomial. In some examples, generation component 535 may be configured or otherwise supported for means of generating error estimates at least in part based on a scaling factor and the difference measurements, the scaling factor being based on an extracted Zelnik factor associated with the Zelnik polynomial.

[0099] In some examples, the patterning component 540 may be configured or otherwise support means for patterning semiconductor material at least in part based on modifying the positioning of the patterning data on the substrate using flatness measurements generated herein.

[0100] Figure 6 A flowchart illustrating a method 600 for substrate measurement techniques supporting anti-flatness error according to various aspects of this disclosure is shown. The operation of method 600 can be implemented by the substrate measurement system or its components described herein. For example, the operation of method 600 can be performed by reference to... Figures 1 to 5 The described system 100 is used to perform this function. In some examples, the substrate measurement system can execute a set of instructions to control the functional elements of the substrate measurement system to perform the described function. Alternatively, the substrate measurement system can use dedicated hardware to perform aspects of the described function.

[0101] At 605, the method may include positioning the substrate in contact with one or more support points. Operation at 605 can be performed according to examples as disclosed herein. In some examples, aspects of operation at 605 may be provided by reference to [reference needed]. Figure 5 The positioning component 525 described herein performs this action.

[0102] At 610, the method may include measuring a first flatness measurement of the gravity-induced deformation of an indicator substrate via an interferometer, the substrate being in a first orientation relative to the vertical direction. Operation 610 can be performed according to examples as disclosed herein. In some examples, aspects of operation 610 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0103] At 615, the method may include measuring a second flatness measurement of the deformation of an indicator substrate via an interferometer, the substrate being at least partially in a second orientation at 90 degrees relative to a first orientation based on a rotated substrate. Operation 615 can be performed according to examples as disclosed herein. In some examples, aspects of operation 615 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0104] At 620, the method may include measuring a third flatness measurement of the deformation of an indicator substrate via an interferometer, the substrate being at least partially based on a rotating substrate in a third orientation 180 degrees relative to a first orientation. Operation 620 can be performed according to examples as disclosed herein. In some examples, aspects of operation 620 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0105] At 625, the method may include measuring a fourth flatness measurement of the deformation of an indicator substrate via an interferometer, the substrate being at least partially based on a rotating substrate in a fourth orientation of 270 degrees relative to a first orientation. Operation 625 can be performed according to examples as disclosed herein. In some examples, aspects of operation 625 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0106] At 630, the method may include generating a first set of differences, each of the first set of differences being at least partially based on differences between corresponding flatness measurement orientations at 90 degrees relative to another corresponding flatness measurement. The operation at 630 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 630 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0107] At 635, the method may include generating a second set of differences, each of the second set of differences being at least partially based on differences between corresponding flatness measurement orientations at 180 degrees relative to another corresponding flatness measurement orientation. Operation 635 can be performed according to examples as disclosed herein. In some examples, aspects of operation 635 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0108] At 640, the method may include fitting each of the first set of differences to a corresponding even-Zelnique polynomial and fitting each of the second set of differences to a corresponding chizelnique polynomial. The operation at 640 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 640 may be derived from references... Figure 5The described generation component 535 is used to execute this.

[0109] At 645, the method may include generating a fifth flatness of the substrate based at least in part on fitting a first set of differences to corresponding even-Zelnique polynomials and fitting a second set of differences to corresponding chisernik polynomials. Operation 645 can be performed according to examples as disclosed herein. In some examples, aspects of operation 645 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0110] At 650, the method may include selecting a flatness measurement value from a first flatness measurement value, a second flatness measurement value, a third flatness measurement value, or a fourth flatness measurement value. The operation at 650 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 650 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0111] At 655, the method may include replacing one or both of the even Zelnique polynomial components and the chizelnique polynomial components associated with the selected flatness measurement with a fifth flatness measurement. The operation at 655 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 655 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0112] Figure 7 A flowchart illustrating a method 700 for substrate measurement techniques supporting resistance to flatness errors according to various aspects of this disclosure is shown. The operation of method 700 can be implemented by the substrate measurement system or its components described herein. For example, the operation of method 700 can be performed using reference to... Figure 1-5 The described system 100 is used to perform this function. In some examples, the substrate measurement system can execute a set of instructions to control the functional elements of the substrate measurement system to perform the described function. Alternatively, the substrate measurement system can use dedicated hardware to perform aspects of the described function.

[0113] At 705, the method may include positioning the substrate in contact with one or more support points. Operation at 705 can be performed according to examples as disclosed herein. In some examples, aspects of operation at 705 may be provided by reference to [reference needed]. Figure 5 The positioning component 525 described herein performs this action.

[0114] At 710, the method may include measuring a first flatness of an indicator substrate of a substrate under gravity-induced deformation via an interferometer, the substrate being in a first orientation relative to the vertical direction. Operation 710 can be performed according to examples as disclosed herein. In some examples, aspects of operation 710 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0115] At 715, the method may include measuring a second flatness of an indicator substrate via an interferometer, the substrate being at least partially in a second orientation at 90 degrees relative to a first orientation based on a rotated substrate. Operation 715 can be performed according to examples as disclosed herein. In some examples, aspects of operation 715 may be derived from references... Figure 5 The described measuring component 530 is used to perform this measurement.

[0116] At 720, the method may include generating a difference measurement between a first flatness measurement and a second flatness measurement. The operation at 720 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 720 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0117] At 725, the method may include fitting the difference measurement to a first corresponding even Zelnik polynomial. The operation at 725 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 725 may be derived from, as referenced... Figure 5 The described generation component 535 is used to execute this.

[0118] At 730, the method may include selecting a smoothness measurement from a first smoothness measurement or a second smoothness measurement, and fitting the selected smoothness measurement to a second corresponding Evenzernik polynomial. The operation at 730 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 730 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0119] At 735, the method may include generating an error estimate based at least in part on subtracting the first Evenzernik polynomial from the second Evenzernik polynomial. The operation at 735 can be performed according to examples as disclosed herein. In some examples, aspects of the operation at 735 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0120] At 740, the method may include generating a third flatness measurement of the substrate based at least in part on an error estimate. Operation 740 can be performed according to examples as disclosed herein. In some examples, aspects of operation 740 may be derived from references... Figure 5 The described generation component 535 is used to execute this.

[0121] In some examples, the device as described herein can perform one or more methods, such as method 600 and / or method 700.

[0122] According to a first aspect, a method for determining the flatness of a substrate, the method comprising: measuring a first flatness measurement of the substrate in a first orientation relative to a vertical direction, measuring a second flatness measurement of the substrate in a second orientation relative to the vertical direction, measuring a third flatness measurement of the substrate in a third orientation relative to the vertical direction, and measuring a fourth flatness measurement of the substrate in a fourth orientation relative to the vertical direction, each of the first orientation, the second orientation, the third orientation, and the fourth orientation being in a different orientation relative to the vertical direction. The method further includes: generating a first set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements; fitting the first set of differences to a corresponding orthogonal polynomial to generate a fifth flatness measurement; selecting a flatness measurement from the first, second, third, and fourth flatness measurements; replacing the orthogonal polynomial component associated with the selected flatness measurement with the orthogonal polynomial component associated with the fifth flatness measurement; and generating the true flatness of the substrate.

[0123] According to the second aspect, the method of the first aspect, wherein the orthogonal polynomial comprises Zernike polynomials.

[0124] According to the third aspect, the method of the second aspect, wherein the step of fitting the first set of differences to the corresponding orthogonal polynomial includes fitting the first set of differences to the corresponding even Zernike polynomial.

[0125] According to the fourth aspect, the method of the first or second aspect, wherein the second orientation is 90 degrees relative to the first orientation, the third orientation is 90 degrees relative to the second orientation, and the fourth orientation is 90 degrees relative to the third orientation.

[0126] According to the fifth aspect, the method described in the first to fourth aspects, wherein generating the first set of differences comprises subtracting one of the first, second, third, and fourth flatness measurements from another of the first, second, third, and fourth flatness measurements, such that the subtracted flatness measurements are oriented at 90 degrees relative to each other.

[0127] According to the sixth aspect, the method of the fifth aspect further includes generating a second set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements.

[0128] According to the seventh aspect, the method of the sixth aspect, wherein generating the second set of differences comprises subtracting one of the first, second, third, and fourth flatness measurements from another of the first, second, third, and fourth flatness measurements, such that the subtracted flatness measurements are oriented 180 degrees relative to each other.

[0129] According to the method of the sixth aspect, in the eighth aspect, wherein the orthogonal polynomial includes a Zelnik polynomial, and the method further includes fitting the second set of differences to the corresponding Zelnik polynomial to generate the fifth flatness measurement.

[0130] According to the ninth aspect, the method described in the first to eighth aspects further includes replacing the orthogonal polynomial component associated with the selected flatness measurement value with the fifth flatness measurement value to generate a sixth flatness measurement value; and subtracting the selected flatness measurement value from the sixth flatness measurement value to generate a seventh flatness measurement value.

[0131] According to the tenth aspect, the method of the ninth aspect, wherein the orthogonal polynomial comprises a Zernike polynomial, and the method further comprises extracting a gravity-related Zernike astigmatism factor from the seventh flatness, generating a gravity-related Zernike defocus factor by multiplying the Zernike astigmatism factor by a scaling factor, and subtracting the Zernike defocus factor from the sixth flatness measurement to generate the true flatness of the substrate.

[0132] According to the eleventh aspect, the method described in the first to tenth aspects further includes selecting a second flatness measurement value from the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value, and replacing the orthogonal polynomial component associated with the selected second flatness measurement value with an orthogonal polynomial component associated with the fifth flatness measurement value.

[0133] According to the twelfth aspect, the method of the first to eleventh aspects, wherein generating the first set of differences includes subtracting the first flatness measurement value from the second flatness measurement value, subtracting the second flatness measurement value from the third flatness measurement value, subtracting the third flatness measurement value from the fourth flatness measurement value, and subtracting the fourth flatness measurement value from the first flatness measurement value.

[0134] According to the thirteenth aspect, the method of the first to twelfth aspects, wherein generating the first set of differences includes subtracting the first flatness measurement value from the second flatness measurement value, subtracting the second flatness measurement value from the third flatness measurement value, subtracting the third flatness measurement value from the fourth flatness measurement value, and subtracting the fourth flatness measurement value from the first flatness measurement value, and the method further includes generating a second set of differences by subtracting the first flatness measurement value from the third flatness measurement value and subtracting the second flatness measurement value from the fourth flatness measurement value.

[0135] According to the fourteenth aspect, the method of the first to thirteenth aspects further includes rotating the substrate to each of the first orientation, the second orientation, the third orientation, and the fourth orientation by removing the substrate from the support and reattaching the substrate to the support.

[0136] According to a fifteenth aspect, an apparatus includes a support configured to hold a substrate in one or more orientations using one or more support members; and an interferometer configured to measure a first flatness measurement of the substrate in a first orientation relative to a vertical direction, a second flatness measurement of the substrate in a second orientation relative to the vertical direction, a third flatness measurement of the substrate in a third orientation relative to the vertical direction, and a fourth flatness measurement of the substrate in a fourth orientation relative to the vertical direction, each of the first orientation, the second orientation, the third orientation, and the fourth orientation being in a different orientation relative to the vertical direction. The device further includes a controller configured to generate a first set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements; fit the first set of differences to a corresponding orthogonal polynomial to generate a fifth flatness measurement; select a flatness measurement from the first, second, third, and fourth flatness measurements; replace an orthogonal polynomial component associated with the selected flatness measurement with an orthogonal polynomial component associated with the fifth flatness measurement; and generate the true flatness of the substrate.

[0137] According to the sixteenth aspect, the device of the fifteenth aspect, wherein the substrate is a photomask.

[0138] According to the seventeenth, fifteenth, or sixteenth aspect of the device, wherein the one or more support members comprise a support member positioned along the top surface of the substrate and a support member positioned along the bottom surface of the substrate.

[0139] According to the eighteenth, fifteenth to seventeenth aspects of the device, wherein each of the one or more support members is a cantilever member extending from the base.

[0140] The device according to the nineteenth aspect, the fifteenth aspect to the eighteenth aspect, further includes a reference surface and one or more optical elements.

[0141] According to the twentieth aspect, the device of the nineteenth aspect, wherein the reference surface is a Fizzo surface.

[0142] According to the twenty-first, fifteenth to twentieth aspects of the device, wherein the second orientation is 90 degrees relative to the first orientation, the third orientation is 90 degrees relative to the second orientation, and the fourth orientation is 90 degrees relative to the third orientation.

[0143] According to a twenty-second aspect, a method includes: measuring a first flatness measurement of the substrate in a first orientation relative to a vertical direction; measuring a second flatness measurement of the substrate in a second orientation relative to the vertical direction; generating a difference measurement between the first flatness measurement and the second flatness measurement; fitting the difference measurement to a corresponding first orthogonal polynomial; selecting a flatness measurement from the first and second flatness measurements, and fitting the selected flatness measurement to a corresponding second orthogonal polynomial; subtracting the first orthogonal polynomial from the second orthogonal polynomial to generate an error estimate; and generating the true flatness of the photomask based at least in part on the error estimate.

[0144] According to the methods of the twenty-third and twenty-second aspects, the first orthogonal polynomial and the second orthogonal polynomial comprise Zernike polynomials.

[0145] According to the twenty-fourth aspect, the method of the twenty-third aspect, wherein the step of fitting the first set of differences to the corresponding orthogonal polynomial includes fitting the first set of differences to the corresponding first even-Zelnique polynomial, and the step of fitting the second set of differences to the corresponding orthogonal polynomial includes fitting the second set of differences to the corresponding second Qizelnique polynomial.

[0146] According to the methods of aspects 25, 22 to 24, wherein the second orientation is at a 90-degree angle relative to the first orientation.

[0147] The method according to the twenty-sixth, twenty-second to twenty-fifth aspects further includes rotating the substrate to each of the first orientation and the second orientation.

[0148] According to the method of aspects 27, 22 to 26, it further comprises scaling the gravity error prediction model to generate the error estimate, and subtracting the error estimate from the selected flatness measurement to generate the true flatness of the photomask.

[0149] It should be noted that these methods describe examples of implementation schemes, and the operations and steps can be rearranged or otherwise modified to make other implementation schemes possible. In some examples, aspects from two or more of the methods described may be combined. For example, each aspect of the methods may include steps or aspects of other methods, or other steps or techniques described herein. Therefore, aspects of this disclosure can provide consumer preferences and maintenance interfaces.

[0150] The description set forth herein, taken in conjunction with the accompanying drawings, illustrates exemplary configurations and does not represent all examples that may be practiced or fall within the scope of the claims. The term "exemplary" as used herein means "serving as an example, instance, or illustration," and not "preferred" or "superior" to other examples. The detailed description includes specific details for the purpose of providing an understanding of the described techniques. However, these techniques may be practiced without these specific details. In some instances, well-known structures and apparatuses are shown in block diagram form to avoid obscuring the concepts of the described examples.

[0151] The various illustrative blocks and modules described in connection with the disclosure herein may be implemented or executed using a general-purpose processor, digital signal processor (DSP), ASIC, field-programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof, designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but alternatively, the processor may be any conventional processor, controller, microcontroller, or state machine. The processor may also be implemented as a combination of computing devices (e.g., a combination of a DSP and a microprocessor, multiple microprocessors, one or more microprocessors combined with a DSP core, or any other such configuration). The function of each unit may also be implemented, in whole or in part, using instructions contained in memory, formatted for execution by one or more general-purpose or special-purpose processors.

[0152] The functionality described herein can be implemented in hardware, software executed by a processor, firmware, or any combination thereof. If implemented in software executed by a processor, the functionality can be stored as one or more instructions or code on or transmitted via a computer-readable medium. Other examples and embodiments are within the scope of this disclosure and the appended claims. For example, due to the nature of software, the functionality described above can be implemented using software executed by a processor, hardware, firmware, hardwired, or any combination thereof. Features implementing the functionality can also be physically located in various locations, including being distributed such that different parts of the functionality are implemented in different physical locations.

[0153] As used herein (including in the claims), the word "or" as used in a list of items (e.g., a list of items beginning with phrases such as "at least one of..." or "one or more of...") indicates an inclusive list, such that a list of at least one of, for example, A, B, or C means A or B or C or AB or AC or BC or ABC (i.e., A and B and C). Furthermore, as used herein, the phrase "based on" should not be construed as referring to a closed set of conditions. For example, an exemplary step described as "based on condition A" may be based on both condition A and condition B without departing from the scope of this disclosure. In other words, as used herein, the phrase "based on" should be interpreted in the same manner as the phrase "at least partially based on".

[0154] In the accompanying drawings, similar parts or features may have the same reference numerals. Additionally, parts of the same type can be distinguished by adding a dash after the reference numeral and a second reference numeral to differentiate similar parts. If only the first reference numeral is used in the description, the description applies to any similar parts that have the same first reference numeral without regard to the second reference numeral.

[0155] The description herein is provided to enable those skilled in the art to make or use this disclosure. Various modifications to this disclosure will be apparent to those skilled in the art, and the general principles defined herein may be applied to other variations without departing from the scope of this disclosure. Therefore, this disclosure is not limited to the examples and designs described herein, but should be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for determining the flatness of a substrate, the method comprising: A first flatness measurement value of the substrate is measured in a first orientation relative to the vertical direction; A second flatness measurement value of the substrate is measured in a second orientation relative to the vertical direction; The third flatness measurement value of the substrate is measured in a third orientation relative to the vertical direction; A fourth flatness measurement value of the substrate is measured in a fourth orientation relative to the vertical direction, wherein each of the first orientation, the second orientation, the third orientation, and the fourth orientation is in a different orientation relative to the vertical direction; Generate a first set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements; The first set of differences is fitted to the corresponding orthogonal polynomial and a fifth flatness measurement value is generated. Select a flatness measurement value from the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value; Replace the orthogonal polynomial component associated with the selected flatness measurement with the orthogonal polynomial component associated with the fifth flatness measurement; as well as The true flatness of the substrate is generated.

2. The method of claim 1, wherein the orthogonal polynomial comprises Zernike polynomial.

3. The method of claim 2, wherein the step of fitting the first set of differences to the corresponding orthogonal polynomial comprises fitting the first set of differences to the corresponding even Zernike polynomial.

4. The method according to claim 1 or claim 2, wherein the second orientation is at 90 degrees relative to the first orientation, the third orientation is at 90 degrees relative to the second orientation, and the fourth orientation is at 90 degrees relative to the third orientation.

5. The method according to any one of claims 1 to 4, wherein generating the first set of differences comprises subtracting one of the first flatness measurement, the second flatness measurement, the third flatness measurement, and the fourth flatness measurement from another of the first flatness measurement, the second flatness measurement, the third flatness measurement, and the fourth flatness measurement, such that the subtracted flatness measurement is oriented at 90 degrees relative to each other.

6. The method of claim 5, further comprising generating a second set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements.

7. The method of claim 6, wherein generating the second set of differences comprises subtracting one of the first, second, third, and fourth flatness measurements from another of the first, second, third, and fourth flatness measurements, such that the subtracted flatness measurements are oriented 180 degrees relative to each other.

8. The method of claim 6, wherein the orthogonal polynomial comprises a Zelnik polynomial, and the method further comprises fitting the second set of differences to a corresponding Zelnik polynomial to generate the fifth flatness measurement.

9. The method according to any one of claims 1 to 8, further comprising: The orthogonal polynomial components associated with the selected flatness measurement are replaced with the fifth flatness measurement to generate a sixth flatness measurement; and The selected flatness measurement value is subtracted from the sixth flatness measurement value to generate the seventh flatness measurement value.

10. The method of claim 9, wherein the orthogonal polynomial comprises a Zernike polynomial, and the method further comprises: Extract the gravity-related Zernike astigmatism factor from the seventh flatness. A gravity-related Zernike power factor is generated by multiplying the Zernike astigmatism factor by a scaling factor; and The true flatness of the substrate is obtained by subtracting the Zernike defocus factor from the sixth flatness measurement.

11. The method according to any one of claims 1 to 10, further comprising: Select the second flatness measurement value from the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value; and Replace the orthogonal polynomial component associated with the selected second flatness measurement with the orthogonal polynomial component associated with the fifth flatness measurement.

12. The method according to any one of claims 1 to 11, wherein generating the first set of differences comprises subtracting the first flatness measurement value from the second flatness measurement value, subtracting the second flatness measurement value from the third flatness measurement value, subtracting the third flatness measurement value from the fourth flatness measurement value, and subtracting the fourth flatness measurement value from the first flatness measurement value.

13. The method according to any one of claims 1 to 12, wherein generating the first set of differences comprises subtracting the first flatness measurement value from the second flatness measurement value, subtracting the second flatness measurement value from the third flatness measurement value, subtracting the third flatness measurement value from the fourth flatness measurement value, and subtracting the fourth flatness measurement value from the first flatness measurement value, and The method further includes generating a second set of differences by subtracting the first flatness measurement value from the third flatness measurement value and subtracting the second flatness measurement value from the fourth flatness measurement value.

14. The method according to any one of claims 1 to 13, further comprising rotating the substrate to each of the first orientation, the second orientation, the third orientation, and the fourth orientation by removing the substrate from the support and reattaching the substrate to the support.

15. An apparatus comprising: A support member configured to hold a substrate in one or more orientations using one or more support members; Interferometer, the interferometer being configured as follows: A first flatness measurement value of the substrate is measured in a first orientation relative to the vertical direction; A second flatness measurement value of the substrate is measured in a second orientation relative to the vertical direction; The third flatness measurement value of the substrate is measured in a third orientation relative to the vertical direction; and A fourth flatness measurement value of the substrate is measured in a fourth orientation relative to the vertical direction, wherein each of the first orientation, the second orientation, the third orientation, and the fourth orientation is in a different orientation relative to the vertical direction; as well as The controller is configured to: Generate a first set of differences between one of the first, second, third, and fourth flatness measurements and another of the first, second, third, and fourth flatness measurements; The first set of differences is fitted to the corresponding orthogonal polynomial and a fifth flatness measurement value is generated. Select a flatness measurement value from the first flatness measurement value, the second flatness measurement value, the third flatness measurement value, and the fourth flatness measurement value; Replace the orthogonal polynomial component associated with the selected flatness measurement with the orthogonal polynomial component associated with the fifth flatness measurement; and The true flatness of the substrate is generated.

16. The device of claim 15, wherein the substrate is a photomask.

17. The device of claim 15 or claim 16, wherein the one or more support members comprise a support member positioned along the top surface of the substrate and a support member positioned along the bottom surface of the substrate.

18. The device according to any one of claims 15 to 17, wherein each of the one or more support members is a cantilever member extending from the base.

19. The device according to any one of claims 15 to 18, further comprising a reference surface and one or more optical elements.

20. The device of claim 19, wherein the reference surface is a Fizeau surface.

21. The device according to any one of claims 15 to 20, wherein the second orientation is 90 degrees relative to the first orientation, the third orientation is 90 degrees relative to the second orientation, and the fourth orientation is 90 degrees relative to the third orientation.

22. A method comprising: A first flatness measurement value of the substrate is measured in a first orientation relative to the vertical direction; A second flatness measurement value of the substrate is measured in a second orientation relative to the vertical direction; Generate a difference measurement value between the first flatness measurement value and the second flatness measurement value; The difference measurement value is fitted to the corresponding first orthogonal polynomial; Select a flatness measurement value from the first flatness and the second flatness measurement values, and fit the selected flatness measurement value to the corresponding second orthogonal polynomial; Subtract the first orthogonal polynomial from the second orthogonal polynomial to generate an error estimate; as well as The true flatness of the photomask is generated based at least in part on the error estimate.

23. The method of claim 22, wherein the first orthogonal polynomial and the second orthogonal polynomial comprise Zernike polynomials.

24. The method of claim 23, wherein the step of fitting the first set of differences to the corresponding orthogonal polynomial comprises fitting the first set of differences to the corresponding first even-Zelnique polynomial, and the step of fitting the second set of differences to the corresponding orthogonal polynomial comprises fitting the second set of differences to the corresponding second Qizelnique polynomial.

25. The method according to any one of claims 22 to 24, wherein the second orientation is at a 90-degree angle relative to the first orientation.

26. The method according to any one of claims 22 to 25, further comprising rotating the substrate to each of the first orientation and the second orientation.

27. The method of any one of claims 22 to 26, further comprising scaling the gravity error prediction model to generate the error estimate, and subtracting the error estimate from the selected flatness measurement to generate the true flatness of the photomask.

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