Lithography system, method and computer program product for hierarchical representation of two-dimensional or three-dimensional shapes

A method of metrology in a metrology apparatus for optimizing a hierarchical description of a shape, the method comprising:

• fitting at least one sub-shape of a first hierarchical order against the shape;
• fitting, after fitting the at least one sub-shape of the first hierarchical order, at least one sub-shape of a second hierarchical order against the shape;
• co-optimizing the at least one sub-shape of the first hierarchical order and the at least one sub-shape of the second hierarchical order;
• and
• storing an optimized hierarchical description of the shape in a memory for use in a manufacturing process,
• wherein each of the at least one sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the at least one sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters.

A method of metrology in a metrology apparatus for modeling experimental data obtained from a shape, the method comprising:

• fitting at least one sub-shape of a first hierarchical order against the experimental data;
• determining data that the fitted at least one sub-shape of the first hierarchical order alone would produce;
• determining residue data, the residue data comprising a difference between the experimental data and the data the fitted at least one sub-shape of the first hierarchical order alone would produce;
• fitting, after the determining residue data step, at least one sub-shape of a second hierarchical order against the residue data;
• determining data that the fitted at least one sub-shape of the second hierarchical order alone would produce;
• combining the data that the fitted at least one sub-shape of the first hierarchical order alone would produce and the data that the fitted at least one sub-shape of the second hierarchical order alone would produce;
• optimizing the fitted at least one sub-shape of the first hierarchical order and the fitted at least one sub-shape of the second hierarchical order based on the combined data and the experimental data;
• determining a hierarchical description of the shape based on the combined data and the experimental data;
• and
• storing the hierarchical description of the shape in a memory for use in a manufacturing process,
• wherein each of the at least one sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the at least one sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters.

A method of metrology in a metrology apparatus for modeling experimental data obtained from a shape, the method comprising:

• fitting at least one first sub-shape of a first hierarchical order against the experimental data;
• fitting, after fitting the at least one first sub-shape of the first hierarchical order, at least one second sub-shape of a second hierarchical order against the experimental data;
• determining data that the fitted at least one sub-shape of the first hierarchical order and the fitted at least one sub-shape of the second hierarchical order jointly would produce;
• optimizing the fitted at least one sub-shape of the first hierarchical order and the fitted at least one sub-shape of the second hierarchical order based on data produced from the first and second sub-shapes and the experimental data,
• determining a hierarchical description of the shape based on data produced from the first and second sub-shapes and the experimental data;
• and
• storing the hierarchical description of the shape in a memory for use in a manufacturing process,
• wherein each of the at least one first sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the first sub-shape of the first hierarchical order,
• wherein each of the at least one second sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the second sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters.

A method of metrology in a metrology apparatus for obtaining a hierarchical description of a shape, the method comprising:

• fitting a first plurality of sub-shapes of a first hierarchical order against the shape;
• determining an error function of the fitting;
• minimizing the error function between the first plurality of sub-shapes of the first hierarchical order and the shape;
• fitting, after the minimizing step, a second plurality of sub-shapes of a second hierarchical order against the error function;
• and
• storing a hierarchical description of the shape in a memory for use in a manufacturing process,
• wherein each of the first plurality of sub-shapes of the first hierarchical order is parameterized as a first minimum number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the second plurality of sub-shapes of the second hierarchical order is parameterized as a second minimum number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first minimum number of geometric parameters is less than the second minimum number of geometric parameters.

A method of metrology in a metrology apparatus for determining one or more parameters of interest of a target of a device pattern on a series of substrates using a patterning process, the method comprising:

• inspecting the target associated with the device pattern, and formed on at least one of the substrates;
• deriving a parameter of interest by:
• fitting at least one sub-shape of a first hierarchical order against a shape;
• determining an error function of the fitting;
• minimizing the error function between the at least one sub-shape of the first hierarchical order and the shape;
• fitting, after the minimizing step, at least one sub-shape of a second hierarchical order against the error function,
• wherein each of the at least one sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the at least one sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters;
• and
• storing the parameter of interest in a memory for use in a manufacturing process.

A computer program product comprising a non-transitory computer readable medium having instructions recorded thereon, the instructions when executed by a computer implement a method comprising:

• fitting a at least one sub-shape of a first hierarchical order against a shape;
• determining an error function of the fitting;
• minimizing the error function between the at least one sub-shape of the first hierarchical order and the shape;
• and
• fitting, after the minimizing step, at least one sub-shape of a second hierarchical order against the error function,
• wherein each of the at least one sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the at least one sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters.

A system comprising:

• an inspection apparatus configured to provide a beam on a measurement target on a substrate and to detect radiation redirected by the target to determine a parameter of a patterning process;
• and
• a non-transitory computer program product configured to evaluate performance of the inspection apparatus in accordance with the parameter of the patterning process, the non-transitory computer program product comprising instructions recorded thereon, the instructions when executed, implement a method comprising:
• fitting at least one sub-shape of a first hierarchical order against a shape of the target;
• determining an error function of the fitting;
• minimizing the error function between the at least one sub-shape of the first hierarchical order and the shape;
• and
• fitting, after the minimizing step, at least one sub-shape of a second hierarchical order against the error function,
• wherein each of the at least one sub-shape of the first hierarchical order is parameterized as a first number of geometric parameters that define the physical shape of the sub-shape of the first hierarchical order,
• wherein each of the at least one sub-shape of the second hierarchical order is parameterized as a second number of geometric parameters that define the physical shape of the sub-shape of the second hierarchical order, and
• wherein the first number of geometric parameters is less than the second number of geometric parameters.