Abstract

This paper describes the Shannon entropy in a partially coherent imaging system with Koehler illumination. Numerical simulation shows that the entropy has a one-to-one relationship with the normalized mutual intensity given by the van Cittert–Zernike theorem. Analytical evaluation shows that the entropy is consistent with the definition of coherence and incoherence, which is also verified by numerical simulations. Additional numerical experiments confirm that the entropy depends on the source intensity distribution, polarization state of the source, object, and pupil. Therefore, the entropy quantitatively measures the degree of coherence of the partially coherent imaging system.

© 2011 Optical Society of America

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  1. L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [Crossref]
  2. J. W. Goodman, Statistical Optics, (Wiley, 2000), Chap. 5.
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.
  4. E. Hecht, Optics, 4th International ed. (Addison Wesley, 2002), Chap. 12.
  5. J. W. Goodman, Statistical Optics, (Wiley, 2000), Chap. 7.
  6. R. E. Swing, “Conditions for microdensitometer linearity,” J. Opt. Soc. Am. 62, 199–207 (1972).
    [Crossref]
  7. R. E. Kinzly, “Partially coherent imaging in a microdensitometer,” J. Opt. Soc. Am. 62, 386–394 (1972).
    [Crossref]
  8. H. Gamo, “Intensity matrix and degree of coherence,” J. Opt. Soc. Am. 47, 976–976 (1957).
    [Crossref]
  9. H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. A 217, 408–432 (1953).
    [Crossref]
  10. 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.
    [Crossref]
  11. J. Perina, Coherence of light (Kluwer, 1985) Chap. 5.
  12. 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]
  13. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol.  III, E.Wolf, ed. (North-Holland, 1964), Chap. 3.
    [Crossref]
  14. E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.
  15. K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Am. A 27, 1311–1321(2010).
    [Crossref]
  16. K. Yamazoe and A. R. Neureuther, “Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil and mask,” Proc. SPIE-Int. Soc. Opt. Eng. 7640, 76400N (2010).
    [Crossref]
  17. For entropy, the term Shannon entropy is used in this paper, although the term von Neumann entropy is used in .
  18. E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Appendix B.
  19. J. Tervo, T. Seta¨la¨, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143(2003).
    [Crossref] [PubMed]
  20. E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources I. Field with a narrow spectral range,” Proc. R. Soc. A 225, 96–111 (1954).
    [Crossref]
  21. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [Crossref]
  22. M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383(1993).
    [Crossref]
  23. I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
    [Crossref]

2010 (2)

K. Yamazoe, “Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients,” J. Opt. Soc. Am. A 27, 1311–1321(2010).
[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-Int. Soc. Opt. Eng. 7640, 76400N (2010).
[Crossref]

2004 (1)

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[Crossref]

2003 (1)

2002 (1)

1993 (1)

1989 (1)

1972 (2)

1965 (1)

L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[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. A 225, 96–111 (1954).
[Crossref]

1953 (1)

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

Born, M.

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

Campos, J.

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[Crossref]

Friberg, A.

Gamo, H.

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

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

Goodman, J. W.

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

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

Hecht, E.

E. Hecht, Optics, 4th International ed. (Addison Wesley, 2002), Chap. 12.

Hopkins, H. H.

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

Kinzly, R. E.

Kutay, M. A.

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.
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

Mansuripur, M.

Moreno, I.

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[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-Int. Soc. Opt. Eng. 7640, 76400N (2010).
[Crossref]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Appendix B.

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

Ozaktas, H. M.

Perina, J.

J. Perina, Coherence of light (Kluwer, 1985) Chap. 5.

Seta¨la¨, T.

Swing, R. E.

Tervo, J.

Vargas, A.

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[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. A 225, 96–111 (1954).
[Crossref]

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

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-Int. Soc. Opt. Eng. 7640, 76400N (2010).
[Crossref]

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

Yüksel, S.

Yzuel, M. J.

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[Crossref]

J. Mod. Opt. (1)

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51, 2031–2038 (2004).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Express (1)

Proc. R. Soc. A (2)

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

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

Proc. SPIE-Int. Soc. Opt. Eng. (1)

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

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

Other (10)

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

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

E. Hecht, Optics, 4th International ed. (Addison Wesley, 2002), Chap. 12.

J. W. Goodman, Statistical Optics, (Wiley, 2000), Chap. 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.
[Crossref]

J. Perina, Coherence of light (Kluwer, 1985) Chap. 5.

For entropy, the term Shannon entropy is used in this paper, although the term von Neumann entropy is used in .

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Appendix B.

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

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

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Figures (11)

Fig. 1
Fig. 1

Partially coherent imaging system assumed in this paper, where FT means Fourier transform relationship. The pupil defines the numerical aperture of the projection optics.

Fig. 2
Fig. 2

Partially coherent source consisting of a pair of point sources represented by Eq. (5). The two point sources are mutually incoherent. The white circle corresponds to the pupil edge.

Fig. 3
Fig. 3

Pair of pinholes whose distance is d is placed on opaque background. The object is represented as a ( x , y ) = rect ( x , y ) + rect ( x d , y ) , where rect ( x , y ) = 1 if     | x | 1 / 2 and | y | 1 / 2 , otherwise 0.

Fig. 4
Fig. 4

(a) Simulation result of the normalized entropy H and the complex degree of coherence | j 12 | by the two-pinhole model in Section 2. Vertical gray lines show the position where | j 12 | = 0 or ± 1 . (b) The relationship between H and | j 12 | when the diffraction effect is negligible.

Fig. 5
Fig. 5

Relationship between the position shift Δ and the PSF ( q ) .

Fig. 6
Fig. 6

Arrangement of three pinholes to confirm the diffraction effect to the normalized entropy H . All pinholes are placed on opaque background. The object is represented as a ( x , y ) = rect ( x , y + 0.2 λ / NA ) rect ( x , y 0.2 λ / NA ) + rect ( x l , y ) .

Fig. 7
Fig. 7

Simulation result of the normalized entropy H and the absolute value of the complex degree of coherence | j 12 | by the three- pinhole model in Section 3C.

Fig. 8
Fig. 8

Difference of the normalized entropy H with and without aberration by the two-pinhole model in Section 2.

Fig. 9
Fig. 9

Examples of object and source used in Section 4. (a) A circular pinhole with radius r = 150 nm ( 0.52 λ / NA ) is placed on opaque background. The object is represented as a ( x , y ) = circ( f 2 + g 2 / r ) . (b)  σ = 0.5 illumination, i. e., S ( f , g ) = circ ( f 2 + g 2 / 0.5 ) . Each pixel is mutually incoherent with all others.

Fig. 10
Fig. 10

Normalized entropy H with various coherence factors and pinhole-size combinations.

Fig. 11
Fig. 11

Normalized entropy H by (a) linearly f-polarized illumination and (b) linearly g-polarized illumination. Vertical gray lines show the positions where | j 12 | of the scalar imaging is either 0 or ± 1 .

Equations (7)

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H = j = 1 N p j log p j ,
I ( x , y ) = ϕ | Z | ϕ ,
I ( x , y ) = j = 1 N | λ j Φ j ( x , y ) | 2 ,
H = j = 1 N ( | λ j | 2 Tr [ Z ] ) log ( | λ j | 2 Tr [ Z ] ) .
S ( f , g ) = δ ( f 1 , 0 ) + δ ( f 2 , 0 ) .
j 12 ( x 1 x 2 ) = 1 2 { exp [ i 2 π NA λ f 1 ( x 1 x 2 ) ] + exp [ i 2 π NA λ f 2 ( x 1 x 2 ) ] } .
j 12 ( d ) = 1 2 [ exp ( i 2 π NA λ f 1 d ) + exp ( i 2 π NA λ f 2 d ) ] .

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