Harmonic resist model for use in a lithographic apparatus and a device manufacturing method

1. A method implemented by a computer, the method being for simulating an image of a mask pattern expected to be produced using a photolithographic process in a resist coated on a substrate, the method comprising:

determining an aerial image of the mask pattern at a level of the substrate; and

convolving, using a processor of the computer, the aerial image with at least two mathematical orthogonal convolution kernels to determine a resist image that is representative of the expected image of the mask pattern in the resist,

wherein the convolution kernels are orthogonal solutions of a two dimensional quantum harmonic oscillator.

2. The method of claim 1 , wherein the resist image is represented by the following equation:

R

=

∑

i

=

0

∞

⁢

c

i

⁡

(

A

*

K

i

)

+

∑

i

=

0

∞

⁢

d

i

⁡

(

A

*

K

1

⁢

i

)

·

(

A

*

K

2

⁢

i

)

wherein R is a bitmap resist image, A is a bitmap aerial image of the mask pattern at the substrate level, K i , K 1i , K 2i are orthogonal convolution kernels and c i and d i are fitting coefficients, where 0≦i≦∞.

3. The method of claim 2 , wherein the fitting coefficient are iteratively adjusted or solved using linear least square methods to reduce a difference between a critical dimension value of the resist image and a critical dimension target value.

4. The method of claim 3 , wherein the fitting coefficients are iteratively adjusted using a Tikhonov-Wahba regression in order to regularize and stabilize the fitting.

5. The method of claim 1 , further comprising converting the resist image into a critical dimension value; comparing the critical dimension value with a target critical dimension value; and adjusting fitting coefficients to reduce a difference between the critical dimension value and the target critical dimension value.

6. The method of claim 1 , wherein the convolving includes convolving the aerial image with a plurality of orthogonal convolution kernels that form a complete basis.

7. The method of claim 1 , wherein the aerial and resist images are two dimensional bitmaps.

8. The method of claim 7 , wherein the convolving includes convolving each pixel of the two dimensional bitmap of the aerial image with a number of orthogonal convolution kernels to determine the two dimensional bitmap of the resist image.

9. The method of claim 1 , wherein a model of the resist image includes a bilinear term that is the product between a first term and a second term, the first term corresponding to the convolution of the aerial image with a first orthogonal convolution kernel and the second term corresponding to the convolution of the aerial image with a second orthogonal convolution kernel.

10. The method of claim 9 , wherein the first and second orthogonal convolution kernels are different.

11. The method of claim 1 , wherein the aerial image corresponds to the image of the mask pattern projected on the substrate through a projection system.

12. The method of claim 1 , wherein the convolving is carried out using FPGA or GPU-based hardware acceleration.

13. A non-transitory computer program product having machine executable instructions, the instructions being executable by a machine to perform a method for simulating an image of a mask pattern expected to be produced using a photolithographic process in a resist coated on a substrate, the method comprising:

determining an aerial image of the mask pattern at a level of the substrate; and

convolving, using a processor of the machine, the aerial image with at least two mathematical orthogonal convolution kernels to determine a resist image that is representative of the expected image of the mask pattern in the resist,

wherein the convolution kernels are orthogonal solutions of a two dimensional quantum harmonic oscillator.

14. The computer program product of claim 13 , wherein the convolving is carried out using a FPGA hardware implementing the processor of the machine.