Abstract

This paper physically compares two different matrix representations of partially coherent imaging with the introduction of matrices E and Z, containing the source, object, and pupil. The matrix E is obtained by extending the Hopkins transmission cross coefficient (TCC) approach such that the pupil function is shifted while the matrix Z is obtained by shifting the object spectrum. The aerial image I can be written as a convex quadratic form I=ϕ|E|ϕ=ϕ|Z|ϕ, where |ϕ is a column vector representing plane waves. It is shown that rank(Z)rank(E)=rank(T)=N, where T is the TCC matrix and N is the number of the point sources for a given unpolarized illumination. Therefore, the matrix Z requires fewer than N eigenfunctions for a complete aerial image formation, while the matrix E or T always requires N eigenfunctions. More importantly, rank(Z) varies depending on the degree of coherence determined by the von Neumann entropy, which is shown to relate to the mutual intensity. For an ideal pinhole as an object, emitting spatially coherent light, only one eigenfunction—i.e., the pupil function—is enough to describe the coherent imaging. In this case, we obtain rank(Z)=1 and the pupil function as the only eigenfunction regardless of the illumination. However, rank(E)=rank(T)=N even when the object is an ideal pinhole. In this sense, aerial image formation with the matrix Z is physically more meaningful than with the matrix E. The matrix Z is decomposed as BB, where B is a singular matrix, suggesting that the matrix B as well as Z is a principal operator characterizing the degree of coherence of the partially coherent imaging.

© 2010 Optical Society of America

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    [CrossRef]
  2. H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.
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    [CrossRef]
  5. E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.
  6. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109–111.
  7. M. J. Bastiaans, “Applications of the Wigner distribution to partially coherent light beams,” in Advances in Information Optics and Photonics, A.T.Friberg and R.Dändliker, eds. (SPIE, 2008), Chap. 2.
    [CrossRef]
  8. W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.
  9. S. B. Mehta and C. J. R. Sheppard, “Phase-space representation of partially coherent imaging systems using the Cohen class distribution,” Opt. Lett. 35, 348–350 (2010).
    [CrossRef] [PubMed]
  10. E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  11. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [CrossRef]
  12. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
    [CrossRef]
  13. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
    [CrossRef]
  14. R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
    [CrossRef]
  15. R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (University of California–Berkeley, 1997).
  16. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 2770–2777 (1982).
    [CrossRef] [PubMed]
  17. R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
    [CrossRef]
  18. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438–2452 (1994).
    [CrossRef]
  19. N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California–Berkeley, 1998).
  20. R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
    [CrossRef]
  21. K. Yamazoe, “Computation theory of partially coherent imaging by stacked pupil shift matrix,” J. Opt. Soc. Am. A 25, 3111–3119 (2008).
    [CrossRef]
  22. R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
    [CrossRef]
  23. K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE 7640, 76400N (2010).
    [CrossRef]
  24. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.
  25. B. Noble and J. W. Daniel, Applied Linear Algebra (Prentice-Hall, 1977), Chap. 5.
  26. H. Gamo, “Intensity matrix and degree of coherence,” J. Opt. Soc. Am. 47, 976 (1957).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 6.
  28. D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
    [CrossRef]
  29. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.
  30. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Field with a narrow spectral range,” Proc. R. Soc. London, Ser. A 225, 96–111 (1954).
    [CrossRef]

2010 (2)

K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE 7640, 76400N (2010).
[CrossRef]

S. B. Mehta and C. J. R. Sheppard, “Phase-space representation of partially coherent imaging systems using the Cohen class distribution,” Opt. Lett. 35, 348–350 (2010).
[CrossRef] [PubMed]

2009 (1)

R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
[CrossRef]

2008 (2)

M. J. Bastiaans, “Applications of the Wigner distribution to partially coherent light beams,” in Advances in Information Optics and Photonics, A.T.Friberg and R.Dändliker, eds. (SPIE, 2008), Chap. 2.
[CrossRef]

K. Yamazoe, “Computation theory of partially coherent imaging by stacked pupil shift matrix,” J. Opt. Soc. Am. A 25, 3111–3119 (2008).
[CrossRef]

2005 (1)

W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.

2004 (1)

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

2003 (1)

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

2002 (1)

1998 (1)

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California–Berkeley, 1998).

1997 (2)

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (University of California–Berkeley, 1997).

1996 (2)

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 6.

1994 (2)

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438–2452 (1994).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

1992 (1)

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

1985 (2)

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109–111.

1983 (1)

1982 (2)

1981 (1)

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1980 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

1977 (1)

B. Noble and J. W. Daniel, Applied Linear Algebra (Prentice-Hall, 1977), Chap. 5.

1964 (1)

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1964), Vol. III, Chap. 3.
[CrossRef]

1957 (1)

1954 (1)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Field with a narrow spectral range,” Proc. R. Soc. London, Ser. A 225, 96–111 (1954).
[CrossRef]

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Barouch, E.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Applications of the Wigner distribution to partially coherent light beams,” in Advances in Information Optics and Photonics, A.T.Friberg and R.Dändliker, eds. (SPIE, 2008), Chap. 2.
[CrossRef]

M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

Cobb, N. B.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California–Berkeley, 1998).

Cole, D. C.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

Conley, W.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Corcoran, N.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Daniel, J. W.

B. Noble and J. W. Daniel, Applied Linear Algebra (Prentice-Hall, 1977), Chap. 5.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

Fung Chen, J.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1964), Vol. III, Chap. 3.
[CrossRef]

H. Gamo, “Intensity matrix and degree of coherence,” J. Opt. Soc. Am. 47, 976 (1957).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 6.

Goodman, W.

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109–111.

Gross, H.

W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.

Hollerbach, U.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

Hsu, S.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Kailath, T.

Kutay, M. A.

Laidig, T.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Mehta, S. B.

Miyakawa, R.

R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
[CrossRef]

Naulleau, P.

R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
[CrossRef]

Neureuther, A. R.

K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE 7640, 76400N (2010).
[CrossRef]

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

Noble, B.

B. Noble and J. W. Daniel, Applied Linear Algebra (Prentice-Hall, 1977), Chap. 5.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

Orszag, S. A.

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

Ozaktas, H. M.

Pati, Y. C.

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438–2452 (1994).
[CrossRef]

Pease, R. F. W.

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

Rabbani, M.

Saleh, B. E. A.

Sheppard, C. J. R.

Shi, X.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Singer, W.

W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.

Socha, R.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Socha, R. J.

R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (University of California–Berkeley, 1997).

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

Starikov, A.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.

Van Den Broeke, D.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

von Bünau, R. M.

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

Wampler, K. E.

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Wang, Y. T.

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

Wolf, E.

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Field with a narrow spectral range,” Proc. R. Soc. London, Ser. A 225, 96–111 (1954).
[CrossRef]

Yamazoe, K.

K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE 7640, 76400N (2010).
[CrossRef]

K. Yamazoe, “Computation theory of partially coherent imaging by stacked pupil shift matrix,” J. Opt. Soc. Am. A 25, 3111–3119 (2008).
[CrossRef]

Yüksel, S.

Zakhor, A.

R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
[CrossRef]

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Vac. Sci. Technol. B (4)

R. M. von Bünau, Y. C. Pati, Y. T. Wang, and R. F. W. Pease, “Optimal coherent decompositions for radially symmetric optical systems,” J. Vac. Sci. Technol. B 15, 2412–2416 (1997).
[CrossRef]

R. Miyakawa, P. Naulleau, and A. Zakhor, “Iterative procedure for in situ extreme ultraviolet optical testing with an incoherent source,” J. Vac. Sci. Technol. B 27, 2927–2930 (2009).
[CrossRef]

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

D. C. Cole, E. Barouch, U. Hollerbach, and S. A. Orszag, “Extending scalar aerial image calculations to higher numerical apertures,” J. Vac. Sci. Technol. B 10, 3037–3041 (1992).
[CrossRef]

Opt. Commun. (1)

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

Opt. Lett. (1)

Physica (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408–432 (1953).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Field with a narrow spectral range,” Proc. R. Soc. London, Ser. A 225, 96–111 (1954).
[CrossRef]

Proc. SPIE (2)

R. Socha, D. Van Den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE 7640, 76400N (2010).
[CrossRef]

Other (12)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1994), Chap. 2.

B. Noble and J. W. Daniel, Applied Linear Algebra (Prentice-Hall, 1977), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 6.

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.

R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (University of California–Berkeley, 1997).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1964), Vol. III, Chap. 3.
[CrossRef]

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109–111.

M. J. Bastiaans, “Applications of the Wigner distribution to partially coherent light beams,” in Advances in Information Optics and Photonics, A.T.Friberg and R.Dändliker, eds. (SPIE, 2008), Chap. 2.
[CrossRef]

W. Singer, M. Totzeck, and H. Gross, “Physical image formation,” in Handbook of Optical Systems, H.Gross, ed. (Wiley, 2005), Vol. 2, Chaps. 19 and 24.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California–Berkeley, 1998).

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Fig. 1
Fig. 1

Schematic diagram to compute the matrix K for one-dimensional M = 7 case. Two point sources are located at f = f 4 = 0 with intensity S 1 and f = f 5 = Δ f with intensity S 2 . The object spectrum is not shifted regardless of the illumination condition. The pupil function is shifted according to the illumination condition. Discretized phasers after the pupils K 1 and K 2 are used to form the matrix K. The first row of the matrix K is K 1 and the second row is K 2 .

Fig. 2
Fig. 2

Schematic diagram to compute the matrix B for one-dimensional M = 7 case. Two point sources are located at f = f 4 = 0 with intensity S 1 and f = f 5 = Δ f with intensity S 2 . The object spectrum is shifted according to the oblique illumination condition. The pupil function is not shifted regardless of the illumination condition. Discretized phasers after the pupils B 1 and B 2 form the matrix B. The first row of the matrix B is B 1 and the second row is B 2 .

Fig. 3
Fig. 3

Schematic view of an object and illumination shape. (a) The target object is an isolated 100 nm square clear aperture placed at opaque background. (b) In the illumination, each pixel is a mutually incoherent point source. The white circle shows ( f 2 + g 2 ) 1 / 2 = 1 .

Fig. 4
Fig. 4

The matrices K and B are both normalized by the same value for the maximum value to be 1. (a) The matrix K whose size is 92 × 3969 . (b) The matrix B whose size is 92 × 961 .

Fig. 5
Fig. 5

Semi-logarithmic plots of the eigenvalues. (a) The eigenvalues of the matrix E as a square of the eigenvalues of the matrix K in Fig. 4a. (b) The eigenvalues of the matrix Z as a square of the eigenvalues of the matrix B in Fig. 4b.

Fig. 6
Fig. 6

Plots of the eigenfunctions. Black circle shows the pupil. (a) The first eigenfunction of the matrix K: Φ 1 . Square sum of every pixel is 1.0000. (b) The first eigenfunction of the matrix B: Φ 1 . Square sum of every pixel is 1.0000.

Fig. 7
Fig. 7

Final aerial images normalized by the same value. (a) Computed as I ( x , y ) = j = 1 92 | F T [ λ j Φ j ( f , g ) ] | 2 . (b) Computed as I ( x , y ) = j = 1 14 | F T [ λ j Φ j ( f , g ) ] | 2 . (c) Difference between (a) and (b).

Fig. 8
Fig. 8

Schematic view of the multi-feature object, which is a combination of clear rectangular apertures placed at opaque background.

Fig. 9
Fig. 9

Semi-logarithmic plots of the eigenvalues calculated from the object shown in Fig. 8 and the illumination shown in Fig. 3b. (a) The eigenvalues of the matrix E. (b) The eigenvalues of the matrix Z.

Fig. 10
Fig. 10

Illumination consists of two point sources. This illumination generates sinusoidal mutual intensity distribution along the X-direction as shown in Fig. 11.

Fig. 11
Fig. 11

The relationship between the mutual intensity from the illumination shown in Fig. 10 and the normalized entropy. Solid line shows the normalized entropy and the dotted line shows the mutual intensity. Chain lines show the points where the mutual intensity is either 0 or ±1.

Equations (32)

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I ( x , y ) = S ( f , g ) | F T [ a ̂ ( f , g ) P ( f + f , g + g ) ] | 2 d f d g ,
( f 1 f 2 f 3 f 4 f 5 f 6 f 7 ) = ( 2 4 3 2 3 0 2 3 4 3 2 ) .
| ϕ 1 D = ( exp ( i 2 π f 1 x ) exp ( i 2 π f 2 x ) exp ( i 2 π f 7 x ) ) T ,
a ̂ 1 1 D = S 1 ( a ̂ 1 a ̂ 2 a ̂ 3 a ̂ 4 a ̂ 5 a ̂ 6 a ̂ 7 ) .
K 1 1 D = S 1 ( 0 0 a ̂ 3 a ̂ 4 a ̂ 5 0 0 ) .
I 1 1 D ( x ) = ϕ 1 D | ( K 1 1 D ) K 1 1 D | ϕ 1 D ,
I 2 1 D ( x ) = ϕ 1 D | ( K 2 1 D ) K 2 1 D | ϕ 1 D ,
K 2 1 D = S 2 ( 0 a ̂ 2 a ̂ 3 a ̂ 4 0 0 0 ) .
I 1 D ( x ) = I 1 1 D ( x ) + I 2 1 D ( x ) = ϕ 1 D | ( K 1 D ) K 1 D | ϕ 1 D ,
K 1 D = ( K 1 1 D K 2 1 D ) .
K 1 D = ( S 1 0 0 S 2 ) ( 0 0 1 1 1 0 0 0 1 1 1 0 0 0 ) ( a ̂ 1 0 0 0 a ̂ 2 0 0 0 a ̂ 7 ) .
ϕ ( f , g ) = ( exp [ i 2 π ( f 1 x + g 1 y ) ] exp [ i 2 π ( f 1 x + g 2 y ) ] exp [ i 2 π ( f 1 x + g M y ) ] exp [ i 2 π ( f 2 x + g 1 y ) ] exp [ i 2 π ( f 2 x + g 2 y ) ] exp [ i 2 π ( f 2 x + g M y ) ] exp [ i 2 π ( f M x + g 1 y ) ] exp [ i 2 π ( f M x + g 2 y ) ] exp [ i 2 π ( f M x + g M y ) ] ) .
| ϕ = Y [ ϕ ( f , g ) ] .
A = ( a ̂ 1 0 a ̂ 2 0 a ̂ M 2 ) .
= ( Y [ P ( f + f 1 , g + g 1 ) ] T Y [ P ( f + f 2 , g + g 2 ) ] T Y [ P ( f + f N , g + g N ) ] T ) ,
S = ( S 1 0 S 2 0 S N ) .
I ( x , y ) = ϕ | ( S A ) S A | ϕ = ϕ | K K | ϕ = ϕ | E | ϕ ,
I ( x , y ) = ϕ | K K | ϕ = Φ | Λ 2 | Φ = j = 1 N | F T [ λ j Φ j ( f , g ) ] | 2 ,
rank ( E ) = min [ rank ( S ) , rank ( ) , rank ( A ) ] = min [ N , N , M 2 ] = N .
rank ( E ) = min [ rank ( S ) , rank ( ) , rank ( A ) ] = min [ 3 N , 3 N , M 2 ] = 3 N .
I ( x , y ) = S ( f , g ) | F T [ a ̂ ( f f , g g ) P ( f , g ) ] | 2 d f d g .
I ( x , y ) = ϕ | ( S A ̃ P ) S A ̃ P | ϕ = ϕ | B B | ϕ = ϕ | Z | ϕ ,
A ̃ = ( Y [ a ̂ ( f f 1 , g g 1 ) ] T Y [ a ̂ ( f f 2 , g g 2 ) ] T Y [ a ̂ ( f f N , g g N ) ] T ) ,
P = ( P 1 0 P 2 0 P ( M / 2 ) 2 ) .
P ( f , g ) = o ( f , g ) exp [ i 2 π W ( f , g ) ] 1 ν 2 NA 2 ( f 2 + g 2 ) 1 NA 2 ( f 2 + g 2 ) circ ( f 2 + g 2 ) = P ( f , g ) circ ( f 2 + g 2 ) ,
P = ( P 1 0 P 2 0 P ( M / 2 ) 2 ) ( circ 1 0 circ 2 0 circ ( M / 2 ) 2 ) = P C .
A ̃ P = ( Y [ a ̂ ( f f 1 , g g 1 ) P x ( f , g ) ] T Y [ a ̂ ( f f N , g g N ) P x ( f , g ) ] T Y [ a ̂ ( f f 1 , g g 1 ) P y ( f , g ) ] T Y [ a ̂ ( f f N , g g N ) P y ( f , g ) ] T Y [ a ̂ ( f f 1 , g g 1 ) P z ( f , g ) ] T Y [ a ̂ ( f f N , g g N ) P z ( f , g ) ] T ) .
I ( x , y ) = ϕ | B B | ϕ = Φ | Λ 2 | Φ = j = 1 N | F T [ λ j Φ j ( f , g ) ] | 2 ,
rank ( Z ) = min [ rank ( S ) , rank ( A ̃ P ) , rank ( C ) ] = min [ N , N , ( M / 2 ) 2 ] = N ,
rank ( Z ) = min [ rank ( S ) , rank ( A ̃ P ) , rank ( C ) ] = min [ 3 N , N , ( M / 2 ) 2 ] = N ,
H = i = 1 N ( λ i 2 Tr [ Z ] ) log ( λ i 2 Tr [ Z ] ) .
U ( x ) = j = 1 M U ̂ ( f j ) exp ( i 2 π f j x ) = ( U ̂ ( f 1 ) U ̂ ( f 2 ) U ̂ ( f M ) ) ( exp ( i 2 π f 1 x ) exp ( i 2 π f 2 x ) exp ( i 2 π f M x ) ) = U ̂ | ϕ .

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