System and method for calculating aerial image of a spatial light modulator

• US 7,394,584 B2
• Filed: 06/30/2005
• Issued: 07/01/2008
• Est. Priority Date: 04/30/2004
• Status: Expired due to Fees

• Current Assignee: ASML Holding NV
• Original Assignee: ASML Holding NV
• Inventors: Latypov, Azat M.
• Primary Examiner(s): Ben; Loha
• Application Number:
  US11/169,587
• Publication Number:
  US 20050243398A1
• Time in Patent Office: 1,097 Days
• Field of Search: 359/237, 359/290, 359/291, 359/298, 359/573, 359/578, 359/559, 356/458, 356/509, 356/520, 430/5, 430/311, 430/690, 430/697, 355/53, 355/67, 348/148, 348/441
• US Class Current: 359/237
• CPC Class Codes: G03F 7/70283 Mask effects on the imaging process
• CPC Class Codes: G03F 7/70291 Addressable masks, e.g. spatial light modulators [SLMs], digital micro-mirror devices [DMDs] or liquid crystal display [LCD] patterning devices
• CPC Class Codes: G03F 7/70441 Optical proximity correction [OPC]
• CPC Class Codes: G03F 7/705 Modelling or simulating from physical phenomena up to complete wafer processes or whole workflow in wafer productions
• CPC Class Codes: G03F 7/70508 Data handling in all parts of the microlithographic apparatus, e.g. handling pattern data for addressable masks or data transfer to or from different components within the exposure apparatus
• CPC Class Codes: G03F 7/70666 Aerial image, i.e. measuring the image of the patterned exposure light at the image plane of the projection system

Abstract

A method of calculating an aerial image of a spatial light modulator array includes calculating pair-wise interference between pixels of the spatial light modulator array; calculating effective graytones corresponding to modulation states of the pixels; and calculating the aerial image based on the pair-wise interference and the effective graytones. The graytones depend only on the modulation states of the pixels. The pair-wise interference depends only on position variables. The position variables are position in an image plane and position in a plane of a source of electromagnetic radiation. The pair-wise interference can be represented by a matrix of functions. The pair-wise interference can be represented by a four dimensional matrix. The effective graytones are approximated using sinc functions, or using polynomial functions.

Claims

1. A method of calculating an aerial image, comprising:

calculating pair-wise interference between pixels of a spatial light modulator array;

calculating effective graytones corresponding to modulation states of the pixels; and

calculating the aerial image based on the pair-wise interference and the effective graytones.

2. The method of claim 1, wherein the effective graytones depend only on the modulation states of the pixels.

3. The method of claim 1, wherein the pair-wise interference depends only on position variables.

4. The method of claim 3, wherein the position variables are position in an image plane and position in a plane of a source of electromagnetic radiation.

5. The method of claim 1, wherein the pair-wise interference is represented by a matrix of functions.

6. The method of claim 1, wherein the pair-wise interference is represented by a four dimensional matrix.

7. The method of claim 1, wherein the effective graytones are approximated using sinc functions.

8. The method of claim 1, wherein the effective graytones are approximated using polynomial functions.

9. The method of claim 1, wherein the aerial image I(x,α

) is calculated according to

$I<br><br>\left(\underset{\_}{x},\underset{\_}{\mathrm{\alpha <br><br>}}\right)={\int <br><br>}_{\mathrm{source}}<br><br>{<br><br>U^{\left(i\right)}<br><br>\left(\underset{\_}{x},\underset{\_}{\mathrm{\alpha <br><br>}},{\underset{\_}{f}}_s\right)<br><br>}^2<br><br><br><br>d<br><br>{\underset{\_}{f}}_s=\underset{\mathrm{j1},\mathrm{j2}=1}{\overset{N}{\sum <br><br>}}<br><br>\underset{\mathrm{k1},\mathrm{k2}=1}{\overset{n}{\sum <br><br>}}<br><br>a_{\mathrm{k1}}^{*}<br><br>\left({\mathrm{\alpha <br><br>}}_{\mathrm{j1}}\right)<br><br>a_{\mathrm{k2}}<br><br>\left({\mathrm{\alpha <br><br>}}_{\mathrm{j2}}\right)<br><br>C_{\mathrm{j1j2k1k2}}<br><br>\left(\underset{\_}{x}\right)$ $\mathrm{where}$ $C_{\mathrm{j1j2k1k2}}<br><br>\left(\underset{\_}{x}\right)={\int <br><br>}_{\mathrm{source}}<br><br>U_{\mathrm{j1}}^{\left(\mathrm{k1}\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_{\mathrm{j1}},{\underset{\_}{f}}_s\right)<br><br>U_{\mathrm{j2}}^{\left(\mathrm{k2}\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_{\mathrm{j2}},{\underset{\_}{f}}_s\right)<br><br><br><br>d<br><br>{\underset{\_}{f}}_s$

is a pixel interference matrix C for a given illumination mode,

$U^{\left(i\right)}<br><br>\left(\underset{\_}{x},\stackrel{-><br><br>}{\mathrm{\alpha <br><br>}},{\underset{\_}{f}}_s\right)=\underset{j=1}{\overset{N}{\sum <br><br>}}<br><br>\underset{k=1}{\overset{n}{\sum <br><br>}}<br><br>a_k<br><br>\left({\mathrm{\alpha <br><br>}}_j\right)<br><br>U^{\left(i\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_j,{\underset{\_}{f}}_s\right),<br><br>U^{\left(k\right)}<br><br>\left(\underset{\_}{x},{\underset{\_}{f}}_s\right)=F<br><br>\left(P_k<br><br>\left({\underset{\_}{f}}_p,{\underset{\_}{f}}_s\right)\right),$

F is a linear operator, n is the number of basis functions, N is the number of pixels, α

=[α

₁, . . . , α

_N] is a modulation state of the pixels, and P_k(f_p,f_s) are basis functions representing a variation of a field U^(k) as a function of pupil and source coordinates.

10. The method of claim 9, further comprising iteratively calculating the aerial image I(x,α

) using sensitivities according to

$\frac{\partial <br><br>I<br><br>\left(\underset{\_}{x},{\underset{\_}{g}}^{\left(0\right)}\right)}{\partial <br><br>g_{\mathrm{j1}}}=\underset{\mathrm{j2}=1}{\overset{N}{\sum <br><br>}}<br><br>g_{j<br><br><br><br>2}^{\left(0\right)}<br><br>\left(C_{j<br><br><br><br>1,j<br><br><br><br>2}<br><br>\left(\underset{\_}{x}\right)+C_{j<br><br><br><br>2,j<br><br><br><br>1}<br><br>\left(\underset{\_}{x}\right)\right)$

where g=[g₁, . . . , g_n] are the pixel graytones.

11. A system for calculating an aerial image, comprising:

means for calculating pair-wise interference between pixels of a spatial light modulator array;

means for calculating effective graytones corresponding to modulation states of the pixels; and

means for calculating the aerial image based on the pair-wise interference and the effective graytones.

12. The system of claim 11, wherein the effective graytones depend only on the modulation states of the pixels.

13. The system of claim 11, wherein the pair-wise interference depends only on position variables.

14. The system of claim 13, wherein the position variables are position in an image plane and position in a plane of a source of electromagnetic radiation.

15. The system of claim 11, wherein the pair-wise interference is represented by a matrix of functions.

16. The system of claim 11, wherein the pair-wise interference is represented by a four dimensional matrix.

17. The system of claim 11, wherein the effective graytones are approximated using sinc functions.

18. The system of claim 11, wherein the effective graytones are approximated using polynomial functions.

19. The system of claim 11, wherein the aerial image I(x,α

) is calculated according to

$I<br><br>\left(\underset{\_}{x},\underset{\_}{\mathrm{\alpha <br><br>}}\right)={\int <br><br>}_{\mathrm{source}}<br><br>{<br><br>U^{\left(i\right)}<br><br>\left(\underset{\_}{x},\underset{\_}{\mathrm{\alpha <br><br>}},{\underset{\_}{f}}_s\right)<br><br>}^2<br><br><br><br>d<br><br>{\underset{\_}{f}}_s=\underset{j<br><br><br><br>1,j<br><br><br><br>2=1}{\overset{N}{\sum <br><br>}}<br><br>\underset{k<br><br><br><br>1,k<br><br><br><br>2=1}{\overset{n}{\sum <br><br>}}<br><br>a_{\mathrm{k1}}^{*}<br><br>\left({\mathrm{\alpha <br><br>}}_{j<br><br><br><br>1}\right)<br><br>a_{k<br><br><br><br>2}<br><br>\left({\mathrm{\alpha <br><br>}}_{j<br><br><br><br>2}\right)<br><br>C_{j<br><br><br><br>1<br><br>\mathrm{j2}<br><br><br><br>k<br><br><br><br>1<br><br>k<br><br><br><br>2}<br><br>\left(\underset{\_}{x}\right)$ $\mathrm{where}$ $C_{j<br><br><br><br>1<br><br>\mathrm{j2}<br><br><br><br>\mathrm{k1}<br><br><br><br>\mathrm{k2}<br><br>}<br><br>\left(\underset{\_}{x}\right)={\int <br><br>}_{\mathrm{source}}<br><br>U_{j<br><br><br><br>1}^{\left(k<br><br><br><br>1\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_{j<br><br><br><br>1},{\underset{\_}{f}}_s\right)<br><br>U_{j<br><br><br><br>2}^{\left(k<br><br><br><br>2\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_{\mathrm{j2}},{\underset{\_}{f}}_s\right)<br><br><br><br>d<br><br>{\underset{\_}{f}}_s$

is a pixel interference matrix C for a given illumination mode,

$U^{\left(i\right)}<br><br>\left(\underset{\_}{x},\mathrm{\alpha <br><br>},{\underset{\_}{f}}_s\right)=\underset{j=1}{\overset{N}{\sum <br><br>}}<br><br>\underset{k=1}{\overset{n}{\sum <br><br>}}<br><br>a_k<br><br>\left({\mathrm{\alpha <br><br>}}_j\right)<br><br>U^{\left(i\right)}<br><br>\left(\underset{\_}{x}-{\underset{\_}{x}}_j,{\underset{\_}{f}}_s\right),<br><br>U^{\left(k\right)}<br><br>\left(\underset{\_}{x},{\underset{\_}{f}}_s\right)=F<br><br>\left(P_k<br><br>\left({\underset{\_}{f}}_p,{\underset{\_}{f}}_s\right)\right),$

F is a linear operator, n is the number of basis functions, N is the number of pixels, α

=[α

₁, . . . , α

_N] is a modulation state of the pixels, and Pk(fp,f5) are basis functions representing a variation of a field U^(k) as a function of pupil and source coordinates.

20. The system of claim 19, further comprising means for iteratively calculating the aerial image I(x,α

) using sensitivities according to

$\frac{\partial <br><br>I<br><br>\left(\underset{\_}{x},{\underset{\_}{g}}^{\left(0\right)}\right)}{\partial <br><br>g_{j<br><br><br><br>1}}=\underset{j<br><br><br><br>2=1}{\overset{N}{\sum <br><br>}}<br><br>g_{j<br><br><br><br>2}^{\left(0\right)}<br><br>\left(C_{j<br><br><br><br>1,j<br><br><br><br>2}<br><br>\left(\underset{\_}{x}\right)+C_{j<br><br><br><br>2,j<br><br><br><br>1}<br><br>\left(\underset{\_}{x}\right)\right)$

where g=[g₁, . . , g_N] are the pixel graytones.