Abstract

Traditionally, aberration correction in extreme ultraviolet (EUV) projection optics requires the use of multiple lossy mirrors, which results in prohibitively high source power requirements. We analyze a single spherical mirror projection optical system where aberration correction is built into the mask itself, through Inverse Lithography Technology (ILT). By having fewer mirrors, this would reduce the power requirements for EUV lithography. We model a single spherical mirror system with orders of magnitude more spherical aberration than would ever be tolerated in a traditional multiple mirror system. By using ILT, (implemented by an adjoint-based gradient descent optimization algorithm), we design photomasks that successfully print test patterns, in spite of these enormous aberrations. This mathematical method was tested with a 6 plane wave illumination source. Nonetheless, it would have poor power throughput from a totally incoherent source.

© 2014 Optical Society of America

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References

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  1. H. J. Levinson, Principles of Lithography, 3rd ed. (SPIE, 2010).
  2. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT), what is the impact to photomask industry?” Luminescent Technologies, Inc. (2006).
  3. Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
    [Crossref]
  4. P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” Proc. SPIE 6924, 69240X (2008).
    [Crossref]
  5. V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
    [Crossref]
  6. W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
    [Crossref]
  7. R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
    [Crossref]
  8. G. Kim, J. A. Domínguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20(3), 2814–2823 (2012).
    [Crossref] [PubMed]
  9. J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 297–305 (1980).
  10. C. Jacobsen and M. R. Howells, “A technique for projection x-ray lithography using computer-generated holograms,” J. Appl. Phys. 71(6), 2993–3001 (1992).
    [Crossref]
  11. J. A. Domínguez-Caballero, S. Takahashi, S. J. Lee, and G. Barbastathis, “Design and fabrication of computer generated holograms for Fresnel domain lithography” in Digital Holography and Three-Dimensional Imaging, Vancouver Canada April 26–30 (2009).
  12. Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).
  13. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
    [Crossref]
  14. P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
    [Crossref]
  15. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. 32(1), 77–79 (2007).
    [Crossref] [PubMed]
  16. V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express 21(7), 8069–8075 (2013).
    [Crossref] [PubMed]
  17. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29(19), 2288–2290 (2004).
    [Crossref] [PubMed]
  18. O. D. Miller, “Photonic design: from fundamental solar cell physics to computational inverse design,” Ph.D. Thesis, EECS Department, Univ. of California, Berkeley (2012).
  19. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
    [Crossref] [PubMed]
  20. V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
    [Crossref]
  21. M. P. Bendsoe and O. Sigmund, Topology Optimization Theory, Methods and Applications (Springer, 2003).
  22. G. Strang, Computational Science and Engineering (Wellesley-Cambridge, 2007).
  23. S. Krantz, A Guide to Complex Variables (2007).
  24. Y. Borodovsky, “EUV lithography at insertion and beyond,” http://www.euvlitho.com/2012/P1.pdf .
  25. C. Solomon and T. Breckon, Fundamentals of Digital Image Processing: a Practical Approach with Examples in MATLAB, (Wiley-Blackwell, 2011).
  26. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  28. V. N. Mahajan, Aberration Theory Made Simple, 2nd ed. (SPIE, 2011).

2014 (1)

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

2013 (2)

2012 (1)

2011 (1)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

2008 (5)

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” Proc. SPIE 6924, 69240X (2008).
[Crossref]

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

2007 (2)

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett. 32(1), 77–79 (2007).
[Crossref] [PubMed]

2006 (1)

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

2004 (1)

1992 (1)

C. Jacobsen and M. R. Howells, “A technique for projection x-ray lithography using computer-generated holograms,” J. Appl. Phys. 71(6), 2993–3001 (1992).
[Crossref]

1980 (1)

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 297–305 (1980).

Bhargava, S.

Bollepalli, S.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Bollepalli, S. B.

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” Proc. SPIE 6924, 69240X (2008).
[Crossref]

Borodovsky, Y.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Cerrina, F.

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

Chegwidden, S.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Cheng, W.

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Cheng, Y.

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

Davids, P. S.

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” Proc. SPIE 6924, 69240X (2008).
[Crossref]

Domínguez-Caballero, J. A.

Dutton, R. W.

Fan, S.

Farnsworth, J.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Fineup, J. R.

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 297–305 (1980).

Frei, W. R.

Frendberg, E.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Ganapati, V.

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

Howells, M. R.

C. Jacobsen and M. R. Howells, “A technique for projection x-ray lithography using computer-generated holograms,” J. Appl. Phys. 71(6), 2993–3001 (1992).
[Crossref]

Hu, B.

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Isoyan, A.

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

Jacobsen, C.

C. Jacobsen and M. R. Howells, “A technique for projection x-ray lithography using computer-generated holograms,” J. Appl. Phys. 71(6), 2993–3001 (1992).
[Crossref]

Jamieson, A.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Jensen, J. S.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

Johnson, H. T.

Khan, M.

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

Kim, G.

Kim, J.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Krantz, S.

S. Krantz, A Guide to Complex Variables (2007).

Kwok, W.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Lalau-Keraly, C. M.

Levi, A. F. J.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

Liu, V.

Liu, Y.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Mahvash, M.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

Menon, R.

Miller, O. D.

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
[Crossref] [PubMed]

Schenker, R.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Seliger, P.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

Sigmund, O.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

Singh, V.

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Toh, K.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Tortorelli, D. A.

Vernon, M.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Veronis, G.

Wagner, S.

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

Wallace, J.

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

Wang, C.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

Wilcox, N.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

Yablonovitch, E.

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
[Crossref] [PubMed]

Yung, K.

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cheng, A. Isoyan, J. Wallace, M. Khan, and F. Cerrina, “Extreme ultraviolet holographic lithography: initial results,” Appl. Phys. Lett. 90023116 (2007).

IEEE J. Photovolt. (1)

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

J. Appl. Phys. (2)

C. Jacobsen and M. R. Howells, “A technique for projection x-ray lithography using computer-generated holograms,” J. Appl. Phys. 71(6), 2993–3001 (1992).
[Crossref]

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(3), 034310 (2006).
[Crossref]

Laser Photon. Rev. (1)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

Opt. Eng. (1)

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 297–305 (1980).

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (5)

Y. Borodovsky, W. Cheng, R. Schenker, and V. Singh, “Pixelated phase mask as novel lithography RET,” Proc. SPIE 6924, 69240E (2008).
[Crossref]

P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” Proc. SPIE 6924, 69240X (2008).
[Crossref]

V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” Proc. SPIE 6924, 69240S (2008).
[Crossref]

W. Cheng, J. Farnsworth, W. Kwok, A. Jamieson, N. Wilcox, M. Vernon, K. Yung, Y. Liu, J. Kim, E. Frendberg, S. Chegwidden, R. Schenker, and Y. Borodovsky, “Fabrication of defect-free full-field pixelated phase mask,” Proc. SPIE 6924, 69241G (2008).
[Crossref]

R. Schenker, S. Bollepalli, B. Hu, K. Toh, V. Singh, K. Yung, W. Cheng, and Y. Borodovsky, “Integration of pixelated phase masks for full-chip random logic layers,” Proc. SPIE 6924, 69240I (2008).
[Crossref]

Other (12)

J. A. Domínguez-Caballero, S. Takahashi, S. J. Lee, and G. Barbastathis, “Design and fabrication of computer generated holograms for Fresnel domain lithography” in Digital Holography and Three-Dimensional Imaging, Vancouver Canada April 26–30 (2009).

H. J. Levinson, Principles of Lithography, 3rd ed. (SPIE, 2010).

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT), what is the impact to photomask industry?” Luminescent Technologies, Inc. (2006).

M. P. Bendsoe and O. Sigmund, Topology Optimization Theory, Methods and Applications (Springer, 2003).

G. Strang, Computational Science and Engineering (Wellesley-Cambridge, 2007).

S. Krantz, A Guide to Complex Variables (2007).

Y. Borodovsky, “EUV lithography at insertion and beyond,” http://www.euvlitho.com/2012/P1.pdf .

C. Solomon and T. Breckon, Fundamentals of Digital Image Processing: a Practical Approach with Examples in MATLAB, (Wiley-Blackwell, 2011).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

V. N. Mahajan, Aberration Theory Made Simple, 2nd ed. (SPIE, 2011).

O. D. Miller, “Photonic design: from fundamental solar cell physics to computational inverse design,” Ph.D. Thesis, EECS Department, Univ. of California, Berkeley (2012).

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Figures (13)

Fig. 1
Fig. 1

A flowchart showing one iteration in the adjoint method. First, electric and/or magnetic fields are found for the current geometry through the forward simulation. Then, the geometry gradient is found through the adjoint simulation. The gradient is used to make an update to the geometry.

Fig. 2
Fig. 2

Projection optics with one mirror (a), and an equivalent system with one lens (b). S is the distance from the mirror to the mask, and S' is the distance from the mirror to the wafer (not to scale). D is the diameter of the mirror/lens.

Fig. 3
Fig. 3

An example Figure-of-Merit calculation at the wafer plane. The color map shows electric field intensity. The desired pattern, Pd, is outlined by the black dashed line. The actual printed pattern, Pa, is outlined in cyan. The “error region”, |PdPa|, is shown in gray. This error region is integrated to obtain the Figure-of-Merit.

Fig. 4
Fig. 4

Illumination pattern. σx = sinθx/NAW and σy = sinθy/NAW. These six plane waves were chosen to give the illumination some of the characteristics of an extended dipole source, such as the one outlined in black. The four σx values are −0.8182, −0.2727, 0.2727, and 0.8182. The three σy values are −0.3099, 0, and 0.3099.

Fig. 5
Fig. 5

(a) mask and (b) wafer plane intensity (normalized to clear field) for an optical system with a naïve mask and no aberrations in the on-axis position. The pattern is periodic, with one unit cell shown. The NA of the system is 0.33, the demagnification is 4, and the wavelength is 13.5 nm.

Fig. 6
Fig. 6

(a) mask and (b) wafer plane intensity (normalized to clear field) for an optical system with a naïve mask in the on-axis position. The pattern is periodic, with one unit cell shown. The NA of the system is 0.33, the demagnification is 4, and the wavelength is 13.5 nm. The mirror radius is 15 cm. The image was taken at the center of the field. This naïve mask is used as the starting geometry for the optimization.

Fig. 7
Fig. 7

(a) Mask and (b) wafer plane intensity (normalized to clear field) for an optical system with an optimized mask. The simulation conditions are the same as in Fig. 6. With this optimized mask, all critical dimensions are within 5% of their target.

Fig. 8
Fig. 8

The same optical system as in Fig. 7, with the mask pixelated. The pixels are 14nm × 15 nm. All critical dimensions are within 8% of their target.

Fig. 9
Fig. 9

(a) Mask and (b) wafer plane intensity (normalized to clear field) for an optical system with a mask optimized to perform through 60nm of defocus. The simulation conditions are the same as in Fig. 5. With this mask, all critical dimensions are within 7% of their target at focus, and remain within 11% through 60nm of defocus.

Fig. 10
Fig. 10

Bossung plots for the worst performing feature for the masks optimized (a) at focus, and (b) for depth-of-focus. For the mask optimized through focus, all critical dimensions remain within 11% of their targets for 60nm of defocus at nominal dose. The sharp jumps seen in the plots correspond to changes in the location of the worst performing feature.

Fig. 11
Fig. 11

A diagram showing the three points on the wafer we designed masks for. The wafer was assumed to be 33 by 26 mm. The mid-field point is displaced 6.5 mm from the optical axis and has >4000 wavelengths of coma and >230 wavelengths of astigmatism (peak value, using the convention in Eq. (31) in Appendix A). The field edge point is displaced 1.3 mm and has >9000 wavelengths of coma, and >900 wavelengths of astigmatism.

Fig. 12
Fig. 12

(a) Mask and (b) wafer plane intensity (normalized to clear field) after optimization for the mid-field location 6.5mm off-axis. The mask resulting from the on-axis optimization was used as the starting mask for this optimization. All critical dimensions are within 2% of their target.

Fig. 13
Fig. 13

(a) Mask and (b) wafer plane intensity (normalized to clear field) after optimization for the field edge location 13mm off-axis. The mask resulting from the mid-field optimization was used as the starting mask for this optimization. All critical dimensions are within 3% of their target.

Equations (31)

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F o M = W ƒ ( E W ( r W ) ) d 2 r W
F o M E M ( r M ) = W ƒ E W ( r W ) E W ( r W ) E M ( r M ) d 2 r W ,
E W ( r W ) = M E M ( r M ) P S F M W ( r W r M ) d 2 r M
F o M E M ( r M ) = W ƒ E W ( r W ) E M ( r M ) [ M E M ( r M ) P S F M W ( r W r M ) d 2 r M ] d 2 r W ,
F o M E M ( r M ) = W ƒ E W ( r W ) [ P S F M W ( r W r M ) ] d 2 r W .
P S F M W ( r W r M ) = P S F W M ( r M r W )
F o M E M ( r M ) = W ƒ E W ( r W ) P S F W M ( r M r W ) d 2 r W .
E M ( r M ) = T M ( r M ) E o e x p [ i ϕ E M ( r M ) ]
F o M T M ( r M ) = F o M E M ( r M ) E M ( r M ) T M ( r M ) + F o M E * M ( r M ) E * M ( r M ) T M ( r M )
F o M T M ( r M ) = 2 R e [ F o M E M ( r M ) E M ( r M ) T M ( r M ) ] .
F o M T M ( r M ) = 2 R e { F o M E M ( r M ) E o e x p [ i ϕ E M ( r M ) ] }
F o M T M ( r M ) = n 2 R e { F o M E M n ( r M ) E o n e x p [ i ϕ E M n ( r M ) ] }
Δ T M F o M T M
F o M = W | P d ( r W ) P a ( E W ( r W ) ) | d 2 r W
P d ( r W ) = { 0 r W desired dark region 1 r W desired bright region P a ( E W ( r W ) ) = { 0 | E W ( r W ) | 2 < I t h 1 | E W ( r W ) | 2 I t h
P a ( E W ( r W ) ) 1 1 + e x p [ A ( | E W ( r W ) | 2 I t h ) ] P a ( E W ( r W ) )
ƒ = | P d P a | = [ ( P d P a ) 2 ] 1 2
f E W = 1 2 [ ( P d P a ) 2 ] 1 2 2 ( P d P a ) ( P a E W )
= P a P d f ( P a E W ) .
P a E W = 1 { 1 + e x p [ A ( | E W | 2 I t h ) ] } 2 E W { e x p [ A ( | E W | 2 I t h ) ] }
= P a 2 E W { e x p [ A ( | E W | 2 I t h ) ] }
= A P a 2 E W * e x p [ A ( | E W | 2 I t h ) ]
f E W = P a P d ƒ A P a 2 E W * e x p [ A ( | E W | 2 I t h ) ] .
E M ( r M ) = T M ( r M ) E o e x p [ i k ( x M sin θ x + y M sin θ y ) ]
F T [ P S F M W ( r M ) ] = O T F ( ρ , ϕ ) = { e x p [ i k O P D ( ρ , ϕ ) ] ρ 1 0 ρ > 1
ρ = ( f x λ N A W ) 2 + ( f y λ N A W ) 2
I W ( r W ) = n | M E M n ( r M ) P S F M W ( r W r M ) d 2 r M | 2
1 S + 1 S = 2 R
N A W = a a 2 + S 2
m = S S .
O P D ( ρ , ϕ ; h ) = a 4 4 R ( 1 R 1 S ) 2 ρ 4 + S R R 2 S 2 a 3 h ρ 3 cos ϕ + a 2 R S 2 h 2 ρ 2 cos 2 ϕ

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