Abstract

A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young’s experiment.

© 2002 Optical Society of America

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  4. J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1971).
  5. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988).
  6. A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1999).
  7. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  8. R. N. Bracewell, “Radio interferometry of discrete sources,” Proc. IRE 46, 97–105 (1958).
    [CrossRef]
  9. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
  10. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, “Sampling and the number of degrees of freedom,” in The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001), Sec. 3.3.
  11. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  12. T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
    [CrossRef]
  13. B. Zhang, B. Lu, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. 27, 99–103 (1996).
    [CrossRef]
  14. The formal analogy between the discrete and continuous cases is discussed in many texts; for instance, see C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977), 2 vols.
  15. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  16. H. Gamo, “Intensity matrix and degree of coherence,” J. Opt. Soc. Am. 47, 976 (1957).
    [CrossRef]
  17. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  18. M. F. Erden, “Repeated filtering in consecutive fractional Fourier domains,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1997).
  19. M. A. Kutay, “Generalized filtering configurations with applications in digital and optical signal and image processing,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1999).
  20. M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).
  21. T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
    [CrossRef]

2001

T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
[CrossRef]

1997

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

1996

B. Zhang, B. Lu, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. 27, 99–103 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

1958

R. N. Bracewell, “Radio interferometry of discrete sources,” Proc. IRE 46, 97–105 (1958).
[CrossRef]

1957

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Bracewell, R. N.

R. N. Bracewell, “Radio interferometry of discrete sources,” Proc. IRE 46, 97–105 (1958).
[CrossRef]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Cohen-Tannoudji, C.

The formal analogy between the discrete and continuous cases is discussed in many texts; for instance, see C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977), 2 vols.

Diu, B.

The formal analogy between the discrete and continuous cases is discussed in many texts; for instance, see C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977), 2 vols.

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

M. F. Erden, “Repeated filtering in consecutive fractional Fourier domains,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1997).

M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).

Ferreira, C.

Friberg, A. T.

T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
[CrossRef]

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

Gamo, H.

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

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Habashy, T.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

Kutay, M. A.

M. A. Kutay, “Generalized filtering configurations with applications in digital and optical signal and image processing,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1999).

M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, “Sampling and the number of degrees of freedom,” in The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001), Sec. 3.3.

Laloë, F.

The formal analogy between the discrete and continuous cases is discussed in many texts; for instance, see C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977), 2 vols.

Lohmann, A. W.

Lu, B.

B. Zhang, B. Lu, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. 27, 99–103 (1996).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Mendlovic, D.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Ozaktas, H. M.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, “Sampling and the number of degrees of freedom,” in The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001), Sec. 3.3.

M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Perina, J.

J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1971).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Visser, T. D.

T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
[CrossRef]

Wolf, E.

T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
[CrossRef]

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Yüksel, S.

M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).

Zalevsky, Z.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, “Sampling and the number of degrees of freedom,” in The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001), Sec. 3.3.

Zhang, B.

B. Zhang, B. Lu, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. 27, 99–103 (1996).
[CrossRef]

Inverse Probl.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

J. Opt.

B. Zhang, B. Lu, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. 27, 99–103 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

T. D. Visser, A. T. Friberg, E. Wolf, “Phase-space inequality for partially coherent optical beams,” Opt. Commun. 187, 1–6 (2001).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Proc. IRE

R. N. Bracewell, “Radio interferometry of discrete sources,” Proc. IRE 46, 97–105 (1958).
[CrossRef]

Other

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, “Sampling and the number of degrees of freedom,” in The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001), Sec. 3.3.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

M. F. Erden, “Repeated filtering in consecutive fractional Fourier domains,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1997).

M. A. Kutay, “Generalized filtering configurations with applications in digital and optical signal and image processing,” Ph.D. thesis (Bilkent University, Ankara, Turkey, 1999).

M. A. Kutay, H. M. Ozaktas, M. F. Erden, S. Yüksel, “Discrete matrix model for synthesis of mutual intensity functions,” in Optical Processing and Computing: A Tribute to Adolf Lohmann, D. P. Casasent, H. J. Caulfield, W. J. Dallas, H. H. Szu, eds., Proc. SPIE4392, 87–98 (2001).

The formal analogy between the discrete and continuous cases is discussed in many texts; for instance, see C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977), 2 vols.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1971).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988).

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete Time Signal Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1999).

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Equations (33)

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Jf=ffH,
Kf=(f-μf)(f-μf)H,
J=UJΛJUJH,
K=UKΛKUKH,
J=k=1NλJkuJkuJkH,
J(m, n)=k=1NλJkuJk(m)uJk*(n),
Jf (x1, x2)=f(x1)f*(x2).
-J(x, x)uν(x)dx=λνuν(x),
--u*(x1)J(x1, x2)u(x2)dx1dx20
-uν*(x)uν(x)dx=δ(ν-ν).
J(x1, x2)=-λ(ν)uν(x1)uν*(x2)dν,
L(m, n)=f(m)f*(n)[|f(m)|2|f(n)|2]1/2=J(m, n)[J(m, m)J(n, n)]1/2,
M(m, n)=[f(m)-μ(m)][f(n)-μ(n)]*[|f(m)-μ(m)|2|f(n)-μ(n)|2]1/2=K(m, n)[K(m, m)K(n, n)]1/2,
L(x1, x2)=f(x1)f*(x2)[|f(x1)|2|f(x2)|2]1/2=J(x1, x2)[J(x1, x1)J(x2, x2)]1/2,
L=I,
|L(m, n)|=1,m, n=1,, N,
J(m, n)=J((m-n) mod N),m, n=1,, N;
L(x1, x2)=δ(x1-x2)/δ(0),
|L(x1, x2)|=1.
c=c-cincohccoher-cincoh,
c1=1Nn=1N(n-1)2λn.
c2=1Nn=1N(λn-1)2,
c3=-n=1NλnNlogλnN,
c4=n=1Nm=1N(m-n)2|L(m, n)|2.
c5=1N2n=1Nm=1N|L(m, n)|2.
c2=2πarctan-λ2(ν)dν,
c3=1πarctan-λ(ν)log λ(ν)dν+12,
Jf˜=Ff(Ff)H=FJfFH.
ΛJ=FJFH.
I=I1+I2+2I1I2|L12|cos[(L12)-α]
=|μ1|2+P1+|μ2|2+P2+2P1P2|μ1μ2*/P1P2+M12|×cos[(μ1μ2*/P1P2+M12)-α],
Visibility=2I1I2|L12|I1+I2
=2P1P2|μ1μ2*/P1P2+M12||μ1|2+P1+|μ2|2+P2.

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