A wafer polishing method includes acquiring in-plane thickness distribution information regarding a wafer to be polished or a wafer subjected to the same processing treatment, determining a difference in pressure between a pressure Pc to be applied to the central part of the wafer by introducing a gas into the central region and a pressure Pe to be applied to the outer peripheral part of the wafer by introducing a gas into the outer peripheral region, determining any one pressure of Pc and Pe, and determining the other pressure, determining the pressure Pg to be applied, based on a set value Pr of a contact pressure to be applied to the lower surface of the second ring-shaped member due to contact with the polishing pad at the time of polishing, and bringing the lower surface of the wafer into contact with the polishing pad to conduct polishing.

Examples are directed to an approach to specifying design guidelines for fabricating a free-form optical component, and methods of fabricating the free-form optical components. In one example, a family of optical prescriptions is created, and a fabricator may manufacture the free-form optical element to any selected prescription within the family. According to certain examples, the series of prescriptions in the family each describe a very similar optical surface providing the same or similar optical performance, and are layered relative to one another such that the thickness of the optical element at any point is slightly less for each subsequent surface defined by the next prescription in the series.

Examples are directed to an approach to specifying design guidelines for fabricating a free-form optical component, and methods of fabricating the free-form optical components. In one example, a family of optical prescriptions is created, and a fabricator may manufacture the free-form optical element to any selected prescription within the family. According to certain examples, the series of prescriptions in the family each describe a very similar optical surface providing the same or similar optical performance, and are layered relative to one another such that the thickness of the optical element at any point is slightly less for each subsequent surface defined by the next prescription in the series.

A double-sided or single-sided machine tool includes a first working disk and a counter-bearing element. The first working disk and the counter-bearing element can be driven rotationally relative to each other by means of a rotary drive. A working gap is formed between the first working disk and the counter-bearing element for the double-sided or single-sided machining of flat workpieces. The double-sided or single-sided machine tool comprises multiple sensors that record measurement data relating to tool and/or machining parameters of the double-sided or single-sided machine tool during operation. A control apparatus obtains the measurement data recorded by the sensors during operation. The control apparatus comprises an artificial neural network that is designed to create a state vector of the double-sided or single-sided machine tool from the measurement data and to compare said state vector with at least one target state vector.

A method for processing a silicon wafer, the method including cutting an ingot to form a wafer, extracting from measured shape data a cross-sectional profile, the cross-sectional profile passing through the center of the wafer and being aligned with a cutting direction of an ingot, interpolating the shape data with a fixed and pre-determined step size, fitting a first second-degree polynomial to the cross-sectional profile, determining a residual profile by subtracting the polynomial from the cross-sectional profile, fitting a second second-degree polynomial to the residual profile using a sliding window of pre-determined width to determine a position, height, and curvature of each peak and valley of the residual profile, determining a waviness parameter based on the position, height, and curvature of each peak and valley of the residual profile, and further processing the wafer based on the waviness parameter and a predetermined waviness threshold.

A method for processing a silicon wafer, the method including cutting an ingot to form a wafer, extracting from measured shape data a cross-sectional profile, the cross-sectional profile passing through the center of the wafer and being aligned with a cutting direction of an ingot, interpolating the shape data with a fixed and pre-determined step size, fitting a first second-degree polynomial to the cross-sectional profile, determining a residual profile by subtracting the polynomial from the cross-sectional profile, fitting a second second-degree polynomial to the residual profile using a sliding window of pre-determined width to determine a position, height, and curvature of each peak and valley of the residual profile, determining a waviness parameter based on the position, height, and curvature of each peak and valley of the residual profile, and further processing the wafer based on the waviness parameter and a predetermined waviness threshold.

The present embodiments provide a mechanism for computing a thickness of a scanned wafer shape to determine a profile, and computing a delta correction value and a polishing end point time by using a computed PV value by the profile and a set predicted PV value and reflecting the same on the polishing time of each wafer which is under polishing. Accordingly, excellent flatness of a wafer surface can be achieved and simultaneously, a plurality of controllers can be controlled simultaneously to reduce equipment cost.

The present embodiments provide a mechanism for computing a thickness of a scanned wafer shape to determine a profile, and computing a delta correction value and a polishing end point time by using a computed PV value by the profile and a set predicted PV value and reflecting the same on the polishing time of each wafer which is under polishing. Accordingly, excellent flatness of a wafer surface can be achieved and simultaneously, a plurality of controllers can be controlled simultaneously to reduce equipment cost.

To specify a trajectory of an eddy current sensor provided on a polishing table of a substrate polishing apparatus, disclosed is a method of identifying a trajectory of an eddy current sensor as seen from a substrate in a substrate polishing apparatus having a polishing table and a polishing head. The method includes: obtaining a sensor output map as three-dimensional data; polishing the substrate; obtaining a profile of the real-time polishing signal as two-dimensional data; and extracting a trajectory having a profile most similar to the profile of the real-time polishing signal as two-dimensional data from the sensor output map as three-dimensional data and identifying the extracted trajectory as a trajectory of the eddy current sensor as seen from the substrate.

To specify a trajectory of an eddy current sensor provided on a polishing table of a substrate polishing apparatus, disclosed is a method of identifying a trajectory of an eddy current sensor as seen from a substrate in a substrate polishing apparatus having a polishing table and a polishing head. The method includes: obtaining a sensor output map as three-dimensional data; polishing the substrate; obtaining a profile of the real-time polishing signal as two-dimensional data; and extracting a trajectory having a profile most similar to the profile of the real-time polishing signal as two-dimensional data from the sensor output map as three-dimensional data and identifying the extracted trajectory as a trajectory of the eddy current sensor as seen from the substrate.