Lithography modeling and applications

A method of building an image representation for use in semiconductor processes, applied to an imaging device that includes a radiation source and an optics device, comprising:

• (a) obtaining a set of source points that collectively represent the radiation source, each source point corresponding to a source region;
• (b) obtaining from a non-transitory computer-readable medium a mask function;
• (c) obtaining a projection function that is dependent upon at least one source point of the set;
• (d) combining the mask function with the projection function to obtain a combined result;
• (e) repeating (c-d) so as to encompass each source point of the set;
• and
• (f) aggregating at least some of the combined results to obtain the image representation;
• where
• the mask function is source-independent, and
• (c) includes performing a convolution of the general form
• ki(x′, y′, z′, z0)m(x′, y′)
• where k represents the projection function, i represents the source point, z′ represents the position within the imaging medium along a dimension of projection, x′ and y′ represent image plane coordinates in a plane substantially perpendicular to the dimension of projection, m(x′,y′) represents the mask function in the image plane coordinate system x′, y′ and z0 represents focal plane position.

A method of building an image representation for use in semiconductor processes, applied to an imaging device that includes a radiation source and an optics device, comprising:

• (a) obtaining a set of source points that collectively represent the radiation source, each source point corresponding to a source region;
• (b) obtaining from a non-transitory computer-readable medium a mask function;
• (c) obtaining a projection function that is dependent upon at least one source point of the set;
• (d) combining the mask function with the projection function to obtain a combined result;
• (e) repeating (c-d) so as to encompass each source point of the set;
• and
• (f) aggregating at least some of the combined results to obtain the image representation;
• where
• obtaining the projection function includes obtaining a pupil function and computing a shifted effective pupil function,
• the effective pupil is shifted by an amount equal to the coordinates of one of the source points, and
• the shifted effective pupil is obtained according to a computation that is of the general form of
• ki,n(x′, y′, z′, z0)=FT−1(Pi,n(α′−αi, β′−βi, z′, z0))
• where FT1 represents the inverse Fourier transform, Pi,n represents the effective pupil function for the coordinate n of the electrical field inside the resist corresponding to the source point i, α′ and β′ are the coordinates in the pupil plane, α and β are the coordinates in the source plane, and αi and βi are the coordinates the source point i in the source plane.

An apparatus comprising instructions stored on non-transitory machine-readable media, the instructions when executed causing a machine to:

• (a) obtain a set of source points that collectively represent a radiation source, each source point corresponding to a source region;
• (b) retrieve from non-transitory computer-readable storage a mask function;
• (c) obtain a projection function dependent upon at least one source point of the set;
• (d) computationally combine the mask function with the projection function to obtain a combined result, and store the combined result in digital memory;
• (e) repeat (c-d) to encompass each source point of the set;
• and
• (f) aggregate at least some of the combined results to obtain a simulated image for the semiconductor mask
• where the instructions further cause a machine when executed to perform optics enhancement by
• performing (f) plural times, each time for a different projection function,
• storing each resultant image representation,
• comparing each of the resultant associated image representations with a semiconductor mask design, and
• selecting a protection function based upon the comparing.

An apparatus comprising instructions stored on non-transitory machine-readable media, the instructions when executed causing a machine to:

• (a) obtain a set of source points that collectively represent a radiation source, each source point corresponding to a source region;
• (b) retrieve from non-transitory computer-readable storage a mask function;
• (c) obtain a projection function dependent upon at least one source point of the set;
• (d) computationally combine the mask function with the projection function to obtain a combined result, and store the combined result in digital memory;
• (e) repeat (c-d) to encompass each source point of the set;
• and
• (f) aggregate at least some of the combined results to obtain a simulated image for the semiconductor mask;
• where the instructions further cause a machine when executed to perform (c) by performing a convolution of the general form
• ki(x′, y′, z′, z0)m(x′, y′)
• where k represents the projection function, i represents the source point, z′ represents the position within the imaging medium along a dimension of projection, x′ and y′ represent image plane coordinates in a plane substantially perpendicular to the dimension of projection, m(x′,y′) represents the mask function in the image plane coordinate system x′, y′ and z0 represents focal plane position.

An apparatus comprising instructions stored on non-transitory machine-readable media, the instructions when executed causing a machine to:

• (a) obtain a set of source points that collectively represent a radiation source, each source point corresponding to a source region;
• (b) retrieve from non-transitory computer-readable storage a mask function;
• (c) obtain a projection function dependent upon at least one source point of the set;
• (d) computationally combine the mask function with the projection function to obtain a combined result, and store the combined result in digital memory;
• (e) repeat (c-d) to encompass each source point of the set;
• and
• (f) aggregate at least some of the combined results to obtain a simulated image for the semiconductor mask;
• where the instructions further cause a machine when executed to
• compute an effective pupil function that is dependent upon shift corresponding to deviation from coordinates of one of the source points, and
• shift the effective pupil according to a computation that is of the general form of
• ki,n(x′, y′, z′, z0)=FT−1(Pi,n(α′−αi, β′−βi, z′, z0))
• where FT1 represents the inverse Fourier transform, Pi,n represents the effective pupil function for the coordinate n of the electrical field inside the resist corresponding to the source point i, α′ and β′ are the coordinates in the pupil plane, α and β are the coordinates in the source plane, and αi and βi are the coordinates the source point i in the source plane.