Alignment modeling and a lithographic apparatus and exposure method using the same

A method comprising:

• determining a position of a first pattern in each of a plurality of target portions on a substrate, based on a fitted mathematical model, wherein the first pattern includes at least one alignment mark, wherein the mathematical model is fitted to a plurality of alignment mark displacements (dx, dy) for the plurality of alignment marks in the target portions, and wherein the alignment mark displacements area difference between a respective nominal position of the alignment mark and measured position of the alignment mark;
• and
• transferring a second pattern onto each of the plurality of target portions, using the determined position of the first pattern in each of the target portions,
• wherein the mathematical model consists essentially of polynomials Z1 and Z2:
• Z1=r2 cos(2θ) Z2=r2 sin(2θ) in polar coordinates (r,θ) or
• Z1=x2−y2 Z2=xy in Cartesian coordinates (x, y).

A computer program product comprising a non-transitory computer readable medium having instructions recorded thereon, the instructions when executed, configured to cause a computer to at least:

• determine a position of a first pattern in each of a plurality of target portions on a substrate, based on a fitted mathematical model, wherein the first pattern includes at least one alignment mark, wherein the mathematical model is fitted to a plurality of alignment mark displacements (dx, dy) for the plurality of alignment marks in the target portions, and wherein the alignment mark displacements are a difference between a respective nominal position of the alignment mark and measured position of the alignment mark;
• and
• cause or have caused transfer of a second pattern onto each of the plurality of target portions, using the determined position of the first pattern in each of the target portions,
• wherein the mathematical model comprises polynomials Z1 and Z2:
• Z1=r2 cos(2θ) Z2=r2 sin(2θ) in polar coordinates (r,θ) or
• Z1=x2−y2 Z2=xy in Cartesian coordinates (x,y), and
• wherein the mathematical model includes a value for each of one or more parameters representing an amplitude and orientation of the polynomials Z1 and Z2.

A computer program product comprising a non-transitory computer readable medium having instructions recorded thereon, the instructions when executed, configured to cause a computer to at least:

• determine a position of a first pattern in each of a plurality of target portions on a substrate, based on a fitted mathematical model, wherein the first pattern includes at least one alignment mark, wherein the mathematical model is fitted to a plurality of alignment mark displacements (dx, dy) for the plurality of alignment marks in the target portions, and wherein the alignment mark displacements are a difference between a respective nominal position of the alignment mark and measured position of the alignment mark;
• and
• cause or have caused transfer of a second pattern onto each of the plurality of target portions, using the determined position of the first pattern in each of the target portions,
• wherein the mathematical model consists essentially of polynomials Z1 and Z2:
• Z1=r2 cos(2θ) Z2=r2 sin(2θ) in polar coordinates (r,θ) or
• Z1=x2−y2 Z2=xy in Cartesian coordinates (x,y).

A computer program product comprising a non-transitory computer readable medium having instructions recorded thereon, the instructions when executed, configured to cause a computer to at least:

• determine a position of a first pattern in each of a plurality of target portions on a substrate, based on a fitted mathematical model, wherein the first pattern includes at least one alignment mark, wherein the mathematical model is fitted to a plurality of alignment mark displacements (dx, dy) for the plurality of alignment marks in the target portions, and wherein the alignment mark displacements are a difference between a respective nominal position of the alignment mark and measured position of the alignment mark;
• and
• cause or have caused transfer of a second pattern onto each of the plurality of target portions, using the determined position of the first pattern in each of the target portions,
• wherein the mathematical model comprises polynomials Z1 and Z2:
• Z1=r2 cos(2θ) Z2=r2 sin(2θ) in polar coordinates (r,θ) or
• Z1=x2−y2 Z2=xy in Cartesian coordinates (x,y), and
• wherein the mathematical model is fitted to the alignment mark displacements (dx, dy) by at least solving:
• (
• dx
• dy
• )
• =
• (
• μ
• ⁢
• ⁢
• x
• ·
• r
• 2
• ⁢
• cos
• ⁡
• (
• 2
• ⁢
• θ
• -
• ax
• )
• μ
• ⁢
• ⁢
• y
• ·
• r
• 2
• ⁢
• sin
• ⁡
• (
• 2
• ⁢
• θ
• -
• ay
• )
• )
• for parameters μx, μy, αx, αy representing an amplitude and radial orientation of the polynomials Z1 and Z2.