Abstract

This paper defines a matrix from which coherence property of imaging by partially coherent Koehler illumination is determined. The matrix termed coherency matrix in imaging system is derived by the space average of a product of a column vector and its transpose conjugate where each row of the column vector represents mutually incoherent light. The coherency matrix in imaging system has similar properties to the polarization matrix that is utilized for calculating the light intensity and degree of polarization of polarized light. The coherency matrix in imaging system enables us to calculate not only image intensity but also degree of coherence for image. Simulation results of the degree of coherence for image given by the coherency matrix in imaging system correspond to the complex degree of coherence obtained by the van Cittert-Zernike theorem.

© 2012 Optical Society of America

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References

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  1. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  2. A. L. Fymat, “Polarization effects in Fourier spectroscopy. I: coherency matrix representation,” Appl. Opt. 11, 160–173 (1972).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2002), Chap. 10.
  4. E. L. O′Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 9.
  5. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), Chap. 8.
  6. R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150 (1977).
    [CrossRef]
  7. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker, eds. (SPIE, 2008), Chap. 9.
  8. K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Amer. A 27, 1311–1321 (2010).
    [CrossRef]
  9. J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 5.
  10. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef]
  11. K. Yamazoe and A. R. Neureuther, “Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence,” J. Opt. Soc. Amer. A 28, 448–454 (2011).
    [CrossRef]
  12. W. Singer, M. Totzeck, and H. Gross, “The Abbe theory of imaging,” in Handbook of Optical Systems, H. Gross, ed. (Wiley-VCH, Darmstadt, 2005), Chap. 21.
  13. A. K. Wong, Optical Imaging in Projection Microlithography, Vol. TT66 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.
  14. R. Barakat, “Partially coherent imaginary in the presence of aberrations,” Optica Acta 17, 337–347 (1970).
    [CrossRef]
  15. H. Gamo, “Intensity matrix and degree of coherence,” J. Opt. Soc. Am. 47, 976–976 (1957).
    [CrossRef]
  16. J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 7.
  17. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Fields with a narrow spectral range,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 225, 96–111 (1954).
    [CrossRef]
  18. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Amer. A 6, 786–805 (1989).
    [CrossRef]
  19. M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Amer. A 10, 382–383 (1993).
    [CrossRef]
  20. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 217, 408–432 (1953).
    [CrossRef]

2011

K. Yamazoe and A. R. Neureuther, “Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence,” J. Opt. Soc. Amer. A 28, 448–454 (2011).
[CrossRef]

2010

K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Amer. A 27, 1311–1321 (2010).
[CrossRef]

2003

1993

M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Amer. A 10, 382–383 (1993).
[CrossRef]

1989

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Amer. A 6, 786–805 (1989).
[CrossRef]

1977

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150 (1977).
[CrossRef]

1972

1970

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

1959

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1957

1954

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

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 217, 408–432 (1953).
[CrossRef]

Barakat, R.

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150 (1977).
[CrossRef]

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

Born, M.

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

Friberg, A.

Fymat, A. L.

Gamo, H.

Goodman, J. W.

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

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

Gross, H.

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

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 217, 408–432 (1953).
[CrossRef]

Luis, A.

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker, eds. (SPIE, 2008), Chap. 9.

Mansuripur, M.

M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Amer. A 10, 382–383 (1993).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Amer. A 6, 786–805 (1989).
[CrossRef]

Neureuther, A. R.

K. Yamazoe and A. R. Neureuther, “Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence,” J. Opt. Soc. Amer. A 28, 448–454 (2011).
[CrossRef]

O'Neill, E. L.

E. L. O′Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 9.

Setälä, T.

Singer, W.

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

Tervo, J.

Totzeck, M.

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

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

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

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), Chap. 8.

Wong, A. K.

A. K. Wong, Optical Imaging in Projection Microlithography, Vol. TT66 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.

Yamazoe, K.

K. Yamazoe and A. R. Neureuther, “Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence,” J. Opt. Soc. Amer. A 28, 448–454 (2011).
[CrossRef]

K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Amer. A 27, 1311–1321 (2010).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Amer. A

K. Yamazoe and A. R. Neureuther, “Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence,” J. Opt. Soc. Amer. A 28, 448–454 (2011).
[CrossRef]

K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Amer. A 27, 1311–1321 (2010).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Amer. A 6, 786–805 (1989).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Amer. A 10, 382–383 (1993).
[CrossRef]

Nuovo Cimento

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Opt. Commun.

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150 (1977).
[CrossRef]

Opt. Express

Optica Acta

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

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.

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

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 217, 408–432 (1953).
[CrossRef]

Other

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

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

A. K. Wong, Optical Imaging in Projection Microlithography, Vol. TT66 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2005), Chap. 4.

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, A. T. Friberg and R. Dändliker, eds. (SPIE, 2008), Chap. 9.

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

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

E. L. O′Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 9.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), Chap. 8.

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Figures (11)

Fig. 1.
Fig. 1.

6f-imaging system assumed in this paper. In the figure, Lc, L, and L are lenses whose focal lengths are fc, f, and f, respectively. The lens Lc is a condenser lens for Koehler illumination. The thick arrows with FT means the Fourier transform. Vector notations x=(x,y) and f=(f,g) are used.

Fig. 2.
Fig. 2.

Partially coherent source consisting of a pair of point sources represented by Eq. (13). The two point sources are mutually incoherent. The white circle shows (f2+g2)1/2=1.

Fig. 3.
Fig. 3.

Pair of ideal pinholes whose distance is d.

Fig. 4.
Fig. 4.

Modified H-S distance p by the illumination in Fig. 2 and the object in Fig. 3. The gray lines show positions where |j12(d)|=0 or 1.

Fig. 5.
Fig. 5.

(a) Simulation result of the modified H-S distance p by the illumination in Fig. 2 and a pair of ideal pinholes located at (0, 6.8736NA/λ) and (d, 0). The gray lines show positions where |j12(d)|=0 or 1; (b) Relationship between p and |j12(d)| when the diffraction effect is negligible.

Fig. 6.
Fig. 6.

Relationship between APSF(q) and the position shift Δ normalized by q.

Fig. 7.
Fig. 7.

Modified H-S distance p by (a) linearly f-polarized illumination and (b) linearly g-polarized illumination. The gray lines show positions where |j12(d)| of unpolarized illumination is either 0 or 1.

Fig. 8.
Fig. 8.

Polarized light just after the pupil when the normal incident coherent illumination is polarized in f-direction and the object is an ideal pinhole. The NA of the imaging optics is set to 0.95 to emphasize the induced polarized electric field. (a) x-polarized electric field a^x(f,g); (b) y-polarized electric field a^y(f,g); (c) z-polarized electric field a^z(f,g).

Fig. 9.
Fig. 9.

Dependency of the modified H-S distance p on NA when normal incident illumination is linearly polarized in f-direction and an object is an ideal pinhole.

Fig. 10.
Fig. 10.

Absolute value of the complex degree of coherence from the origin calculated by the illumination in Fig. 2.

Fig. 11.
Fig. 11.

Comparison of the modified H-S distance of the C matrix and |j12(d)|.

Equations (27)

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I(x,y)=S(f,g)|FT[a^(ff,gg)P(f,g)]|2dfdg,
I(x,y)=i=1NS(fi,gi)|FT[a^(ffi,ggi)P(f,g)]|2=i=1N|Fi(x,y)|2,
Fi(x,y)=S(fi,gi)FT[a^(ffi,ggi)P(f,g)]=S(fi,gi)a^(ffi,ggi)P(f,g)exp[i2π(fx+gy)]dfdg=ei(f,g)exp[i2π(fx+gy)]dfdg.
F=(F1(x,y)F2(x,y)FN(x,y)).
C=FFspace=FFdxdy,
Tr[C]=Tr[FFdxdy]=i=1N|Fi(x,y)|2dxdy=I(x,y)dxdy.
p=NN1C/Tr[C]I/Tr[I]HS=NN1Tr[(C/Tr[C]I/N)(C/Tr[C]I/N)]=NN1[Tr[CC]Tr[C]21N].
Fi(x,y)Fj*(x,y)space=Fi(x,y)Fj*(x,y)dxdy=ei(f,g)ej*(f,g)dfdg,
e=(e1(f,g)e2(f,g)eN(f,g)).
C=eedfdg,
B=(e1(f1,g1)e1(fM,g1)e1(f1,g2)e1(fM,g2)e1(f1,gM)e1(fM,gM)e2(f1,g1)e2(fM,g1)e2(f1,g2)e2(fM,g2)e2(f1,gM)e2(fM,gM)eN(f1,g1)eN(fM,g1)eN(f1,g2)eN(fM,g2)eN(f1,gM)eN(fM,gM)).
C=BB.
S(f,g)=s1δ(ff1,g)+s2δ(ff2,g),
j12(x1x2)=1s1+s2{s1exp[i2πNAλf1(x1x2)]+s2exp[i2πNAλf2(x1x2)]}.
j12(d)=1s1+s2{s1exp[i2πNAλf1d]+s2exp[i2πNAλf2d]}.
I(x,y)=|F1(x,y)|2+|F2(x,y)|2.
I(x,y)=λ1|ϕ1(x,y)|2+λ2|ϕ2(x,y)|2,
ϕi(x,y)ϕj*(x,y)dxdy=δi,j,
I(x,y)=[Γ(x1x2,y1y2)a(x1,y1)a*(x2,y2)K(xx1,yy1)K*(xx2,yy2)]dx1dy1dx2dy2,
Γ(x1x2,y1y2)=S(f,g)exp{i2π[(x1x2)f+(y1y2)g]}dfdg,
K(x,y)=P(f,g)exp[i2π(fx+gy)]dfdg.
a(x,y)=δ(x,y)+δ(xd,y).
I(x,y)=|K(x,y)+K(xd,y)|2.
I(x,y)=i=1Nλi|ϕi(x,y)|2.
TCC(f1,f2,g1,g2)=S(f,g)P(f1+f,g2+g)P*(f2+f,g2+g)]dfdg.
I(x,y)=TCC(f1,f2,g1,g2)a^(f1,g1)a^*(f2,g2)exp{i2π[(f1f2)x+(g1g2)y]}df1df2dg1dg2.
I(x,y)=S(f,g)|FT[a^(f,g)P(f+f,g+g)]|2dfdg.

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