Abstract

This paper discusses the physical difference between two image formation approaches for partially coherent imaging. In one approach, the pupil function is shifted according to illumination condition as in the transmission cross coefficient (TCC) approach, whereas in the other approach, the object spectrum is shifted. Although the two approaches result in identical images, they are built on distinct physical models. Eigenfunction analysis reveals that the TCC approach is built on an artificial optical model only for image calculation. Therefore, the two approaches are not interchangeable except for image calculation. Such an example is found in calculating the entropy in an imaging system.

© 2012 Optical Society of America

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References

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  1. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
    [CrossRef]
  2. J. W. Goodman, Statistical Optics, Wiley Classical Library ed. (Wiley, 2000), Chap. 7.
  3. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
    [CrossRef]
  4. Y. Ichioka and T. Suzuki, “Image of a periodic complex object in an optical system under partially coherent illumination,” J. Opt. Soc. Am. 66, 921–932 (1976).
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  5. B. J. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron. Devices 27, 931–938 (1980).
    [CrossRef]
  6. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. 217, 408–432 (1953).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2002), Chap. 10.
  8. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1964), Vol. III, Chap. 3.
  9. J. Peřina, “Theory of coherence,” Czech. J. Phys. 19, 151–194 (1969).
    [CrossRef]
  10. 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).
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  17. 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).
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    [CrossRef]
  19. K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Am. A 271311–1321 (2010).
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    [CrossRef]

2012 (1)

2011 (1)

2010 (2)

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 (1)

2005 (1)

C. Spence, “Full-chip lithography simulation and design analysis—how OPC is changing IC design,” Proc. SPIE 5751, 1–14 (2005).
[CrossRef]

1996 (1)

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]

1994 (2)

1986 (1)

1982 (1)

1980 (1)

B. J. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron. Devices 27, 931–938 (1980).
[CrossRef]

1976 (1)

1970 (1)

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Optica Acta 17, 337–347 (1970).
[CrossRef]

1969 (1)

J. Peřina, “Theory of coherence,” Czech. J. Phys. 19, 151–194 (1969).
[CrossRef]

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. 217, 408–432 (1953).
[CrossRef]

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[CrossRef]

1938 (1)

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

Barakat, R.

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Optica Acta 17, 337–347 (1970).
[CrossRef]

Bastiaans, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2002), Chap. 10.

Gamo, H.

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

Goodman, J. W.

J. W. Goodman, Statistical Optics, Wiley Classical Library ed. (Wiley, 2000), Chap. 7.

J. W. Goodman, Statistical Optics, Wiley Classical Library ed. (Wiley, 2000), Chap. 5.

Gross, H.

W. Singer, M. Totzeck, and H. Gross, “The Abbe theory of imaging,” in Handbook of Optical Systems 2, H. Gross, ed. (Wiley-VCH, 2005), Chap. 21.

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. 217, 408–432 (1953).
[CrossRef]

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[CrossRef]

Ichioka, Y.

Kailath, T.

Lin, B. J.

B. J. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron. Devices 27, 931–938 (1980).
[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.

O’Neill, E. L.

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

Pati, Y. C.

Perina, J.

J. Peřina, “Theory of coherence,” Czech. J. Phys. 19, 151–194 (1969).
[CrossRef]

Rabbani, M.

Saleh, B. E. A.

Sheppard, C. J. R.

Singer, W.

W. Singer, M. Totzeck, and H. Gross, “The Abbe theory of imaging,” in Handbook of Optical Systems 2, H. Gross, ed. (Wiley-VCH, 2005), Chap. 21.

Socha, R. J.

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]

Spence, C.

C. Spence, “Full-chip lithography simulation and design analysis—how OPC is changing IC design,” Proc. SPIE 5751, 1–14 (2005).
[CrossRef]

Suzuki, T.

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, “The Abbe theory of imaging,” in Handbook of Optical Systems 2, H. Gross, ed. (Wiley-VCH, 2005), Chap. 21.

van der Gracht, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2002), Chap. 10.

Yamazoe, K.

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. (2)

Czech. J. Phys. (1)

J. Peřina, “Theory of coherence,” Czech. J. Phys. 19, 151–194 (1969).
[CrossRef]

IEEE Trans. Electron. Devices (1)

B. J. Lin, “Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication,” IEEE Trans. Electron. Devices 27, 931–938 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Vac. Sci. Technol. B (2)

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]

Opt. Lett. (1)

Optica Acta (1)

R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Optica Acta 17, 337–347 (1970).
[CrossRef]

Physica (1)

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

Proc. Royal Soc. (2)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[CrossRef]

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. Royal Soc. 217, 408–432 (1953).
[CrossRef]

Proc. SPIE (1)

C. Spence, “Full-chip lithography simulation and design analysis—how OPC is changing IC design,” Proc. SPIE 5751, 1–14 (2005).
[CrossRef]

Other (6)

J. W. Goodman, Statistical Optics, Wiley Classical Library ed. (Wiley, 2000), Chap. 7.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2002), Chap. 10.

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

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

W. Singer, M. Totzeck, and H. Gross, “The Abbe theory of imaging,” in Handbook of Optical Systems 2, H. Gross, ed. (Wiley-VCH, 2005), Chap. 21.

J. W. Goodman, Statistical Optics, Wiley Classical Library ed. (Wiley, 2000), Chap. 5.

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Figures (8)

Fig. 1.
Fig. 1.

6f-imaging system configuration. Source and pupil planes as well as object and image planes are under conjugate relationship. In the figure, Lc, L, and L are lenses and the corresponding focal lengths are l, l, and l.

Fig. 2.
Fig. 2.

Rough sketches for (a) Eq. (3) and (b) Eq. (4).

Fig. 3.
Fig. 3.

Partially coherent illumination used for the simulation in Section 3. The white circle shows (f2+g2)1/2=1. Each white pixel shows a point source with unit intensity that is spatially coherent but mutually incoherent with all others.

Fig. 4.
Fig. 4.

(a) Image obtained by Eq. (5) (the pupil shift method). (b) Image obtained by Eq. (6) (the spectrum shift method). (c) The difference between (a) and (b).

Fig. 5.
Fig. 5.

Eigenvalues of the spectrum shift method (marked “○”) and the pupil shift method (marked “×”). The square sum of all eigenvalues is normalized to unity.

Fig. 6.
Fig. 6.

First eigenfunctions by the illumination in Fig. 3 and an ideal pinhole. The white circle shows the pupil edge: (a) Φ1(f,g) and (b) Φ1(f,g).

Fig. 7.
Fig. 7.

Simulation condition to calculate the entropy. (a) The effective source. (b) A pair of ideal pinholes whose distance in the x direction is d. The ideal pinholes are enlarged for visualization.

Fig. 8.
Fig. 8.

Comparison of the normalized entropy and the complex degree of coherence. The vertical lines show the point where |j12(d)| is 0 or 1.

Equations (17)

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I(x,y)=T(f1,g1,f2,g2)a^(f1,g1)a^*(f2,g2)×exp{i2π[(f1f2)x+(g1g2)y]}df1dg1df2dg2,
T(f1,g1,f2,g2)=S(f,g)P(f+f1,g+g1)P*(f+f2,g+g2)dfdg,
I(x,y)=S(f,g)|FT[a^(f,g)P(f+f,g+g)]|2dfdg,
I(x,y)=S(f,g)|FT[a^(ff,gg)P(f,g)]|2dfdg.
I(x,y)=j=1N|μjFT[Φj(f,g)]|2=j=1N|μjϕj(x,y)|2,
I(x,y)=j=1N|μjFT[Φj(f,g)]|2=j=1N|μjϕj(x,y)|2,
I(x,y)=j=1NS(fj,gj)|a^(ffj,ggj)P(f,g)exp[i2π(fx+gy)]dfdg|2=j=1N|Ej(x,y)|2,
ϕi*(x,y)ϕj(x,y)dxdy=δij.
j=1Nμj2=I(x,y)dxdy.
a^(f,g)=FT[δ(x,y)]=1.
I(x,y)=j=1NS(fj,gj)|FT[P(f,g)]|2=|FT[P(f,g)]|2j=1NS(fj,gj)=|j=1NS(fj,gj)FT[P(f,g)]|2=|E(x,y)|2.
I(x,y)=j=1NS(fj,gj)|FT[P(f+fj,g+gj)]|2=j=1N|S(fj,gj)FT[P(f+fj,g+gj)]|2=j=1N|Ej(x,y)|2.
I(x,y)=j=1NS(fj,gj)|a^(f,g)P(f+fj,g+gj)exp[i2π(fx+gy)]dfdg|2.
H=1logNi=1Mρi2logρi2.
S(f,g)=s1δ(ff1,g)+s2δ(ff2,g),
j12(x1x2)=1s1+s2{s1exp[i2πNAλf1(x1x2)]+s2exp[i2πNAλf2(x1x2)]}.
j12(d)=12[exp(i2πNAλ1631d)+exp(i2πNAλ2431d)].

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