Abstract

What is the maximum visibility attainable in double-slit interference by an electromagnetic field if arbitrary – but reversible – polarization and spatial transformations are applied? Previous attempts at answering this question for electromagnetic fields have emphasized maximizing the visibility under local polarization transformations. I provide a definitive answer in the general setting of partially coherent electromagnetic fields. An analytical formula is derived proving that the maximum visibility is determined by only the two smallest eigenvalues of the 4×4 two-point coherency matrix associated with the electromagnetic field. This answer reveals, for example, that any two points in a spatially incoherent scalar field can always achieve full interference visibility by applying an appropriate reversible transformation spanning both the polarization and spatial degrees of freedom – without loss of energy. Surprisingly, almost all current measures predict zero-visibility for such fields. This counter-intuitive result exploits the higher dimensionality of the Hilbert space associated with vector – rather than scalar – fields to enable coherency conversion between the field’s degrees of freedom.

© 2017 Optical Society of America

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References

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    [Crossref]
  9. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [Crossref]
  10. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
    [Crossref] [PubMed]
  13. M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414–2417 (2004).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2015 (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

2014 (3)

2013 (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

2010 (1)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

2007 (5)

2006 (2)

2005 (4)

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express,  13, 6051–6060 (2005).
[Crossref] [PubMed]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

J. Ellis and A. Dogariu, “On the degree of polarization of random electromagnetic fields,” Opt. Commun. 253, 257–265 (2005).
[Crossref]

Y. Aharonov and M. S. Zubairy, “Time and the quantum: Erasing the past and impacting the future,” Science 307875–879 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (2)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

1998 (2)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix”, Pure Appl. Opt. 7, 941–951 (1998).
[Crossref]

1979 (1)

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

1965 (1)

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

1963 (1)

B. Karczewski, “Coherence theory of the electromagnetic field,” Il Nuovo Cimento 30, 5464–5473 (1963).
[Crossref]

1938 (1)

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

1804 (1)

T. Young, “The Bakerian Lecture: Experiments and calculations relative to physical optics,” Phil. Trans. R. Soc. Lond. 94, 1–16 (1804).
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Aharonov, Y.

Y. Aharonov and M. S. Zubairy, “Time and the quantum: Erasing the past and impacting the future,” Science 307875–879 (2005).
[Crossref] [PubMed]

Aiello, A.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Al-Qasimi, A.

Borghi, R.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Byron, F. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, 1992).

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993).

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

J. Ellis and A. Dogariu, “On the degree of polarization of random electromagnetic fields,” Opt. Commun. 253, 257–265 (2005).
[Crossref]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref] [PubMed]

M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414–2417 (2004).
[Crossref]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

J. Ellis and A. Dogariu, “On the degree of polarization of random electromagnetic fields,” Opt. Commun. 253, 257–265 (2005).
[Crossref]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref] [PubMed]

Feynman, R.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics Vol. III (Addison Wesley, 1965).

Friberg, A. T.

Fuller, R. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, 1992).

Gamel, O.

Giacobino, E.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Giuseppe, G. Di

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Gori, F.

Goudail, F.

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix”, Pure Appl. Opt. 7, 941–951 (1998).
[Crossref]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge Univ. Press, 1990).

James, D.

James, D. F. V.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge Univ. Press, 1990).

Kagalwala, K. H.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Karczewski, B.

B. Karczewski, “Coherence theory of the electromagnetic field,” Il Nuovo Cimento 30, 5464–5473 (1963).
[Crossref]

Kipnis, N.

N. Kipnis, History of the Principle of Interference of Light (BirkhäuserVerlag, 1991).
[Crossref]

Korotkova, O.

Leighton, R.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics Vol. III (Addison Wesley, 1965).

Leuchs, G.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Luis, A.

Mandel, L.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
[Crossref]

Marquardt, C.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Martínez-Herrero, R.

Mejías, P. M.

Mujat, M.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Perina, J.

J. Peřina, Coherence of Light (Van Nostrand, 1972).

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

Réfrégier, P.

P. Réfrégier and A. Roueff, “Intrinsic coherence: A new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30–35 (2007).
[Crossref]

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express,  13, 6051–6060 (2005).
[Crossref] [PubMed]

Roueff, A.

P. Réfrégier and A. Roueff, “Intrinsic coherence: A new concept in polarization and coherence theory,” Opt. Photon. News 18(2), 30–35 (2007).
[Crossref]

Saleh, B. E. A.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Sands, M.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics Vol. III (Addison Wesley, 1965).

Santarsiero, M.

Setälä, T.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Tervo, J.

Töppel, F.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix”, Pure Appl. Opt. 7, 941–951 (1998).
[Crossref]

Wolf, E.

A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32, 1015–1016 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[Crossref] [PubMed]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414–2417 (2004).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

Wootters, W. K.

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

W. K. Wootters, “Local accessibility of quantum states,” in Complexity, Entropy, and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, 1990), pp. 39–46.

Young, T.

T. Young, “The Bakerian Lecture: Experiments and calculations relative to physical optics,” Phil. Trans. R. Soc. Lond. 94, 1–16 (1804).
[Crossref]

Zernike, F.

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

Zubairy, M. S.

Y. Aharonov and M. S. Zubairy, “Time and the quantum: Erasing the past and impacting the future,” Science 307875–879 (2005).
[Crossref] [PubMed]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

Il Nuovo Cimento (1)

B. Karczewski, “Coherence theory of the electromagnetic field,” Il Nuovo Cimento 30, 5464–5473 (1963).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photon. (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

New J. Phys. (1)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Opt. Commun. (2)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[Crossref]

J. Ellis and A. Dogariu, “On the degree of polarization of random electromagnetic fields,” Opt. Commun. 253, 257–265 (2005).
[Crossref]

Opt. Express (2)

Opt. Lett. (10)

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
[Crossref] [PubMed]

A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32, 1015–1016 (2007).
[Crossref] [PubMed]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[Crossref] [PubMed]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Global polarization unitary transformation Ûp applied to both r a and r b. (b) Local polarization unitaries U ^ p ( a ) and U ^ p ( b ) applied at r a and r b, respectively.
Fig. 2
Fig. 2 (a) Spatial unitary transformation Ûs that is polarization-independent, depicted as a generalized beam splitter [Eq. 6]. (b) Spatial polarization transformation that is polarization-dependent [Eq. 7], depicted as a polarizing beam splitter. The H and V polarization components undergo different spatial unitaries U ^ s ( H ) and U ^ s ( V ), respectively.
Fig. 3
Fig. 3 General unitary transformation extending across the spatial and polarization DoFs formed as a cascade of the polarization and spatial unitaries in Fig. 1 and Fig. 2, respectively.
Fig. 4
Fig. 4 (a) Two-points in a spatially incoherent scalar field G1 produce no interference. (b) Reversibly transforming the field in (a) to produce full-visibility interference fringes via a succession of unitaries: Û12 rotates the polarization at r b by π 2, G2; Û23 is a polarizing beam splitter that combines the field at r a and r b to produce unpolarized light at r a, G3; and, finally, a non-polarizing beam splitter Û34 produces a spatially coherent – albeit unpolarized – field G4. (c) Reversibly transforming a scalar incoherent field to produce full visibility interference fringes using a sequence of unitaries similar to (b). HWP: half-wave plate; BS: beam splitter; PBS: polarization BS.

Equations (14)

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D s = max { G aa G bb } = V max ,
G = ( G aa HH G aa HV G ab HH G ab HV G aa VH G aa VV G ab VH G ab VV G ba HH G ba HV G bb HH G bb HV G ba VH G ba VV G bb VH G bb VV ) = ( G aa G ab G ba G bb ) ,
G D = ( λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4 ) = diag { λ 1 , λ 2 , λ 3 , λ 4 } ,
G U ^ G U ^ = ( U ^ p G aa U ^ p U ^ p G ab U ^ p U ^ p G ba U ^ p U ^ p G bb U ^ p ) .
G ( U ^ p ( a ) G aa U ^ p ( a ) U ^ p ( a ) G ab U ^ p ( b ) U ^ p ( b ) G ba U ^ p ( a ) U ^ p ( b ) G bb U ^ p ( b ) ) .
U ^ BS = 1 2 ( 1 0 i 0 0 1 0 i i 0 1 0 0 i 0 1 ) .
U ^ PBS = ( 1 0 0 0 0 0 0 i 0 0 1 0 0 i 0 0 ) ,
G s ( r ) = ( G aa HH + G aa VV G ab HH + G ab VV G ba HH + G ba VV G bb HH + G bb VV ) ,
V 0 = 2 | G ab HH + G ab VV | = 2 | Tr { G ab } | ,
V LPU = 2 ( μ 1 + μ 2 ) = 2 Tr { G ab G ab } + 2 | det { G ab } | ,
V max = λ 1 + λ 2 λ 3 λ 4 = 1 2 ( λ 3 + λ 4 ) .
G 1 = 1 10 ( 2 0 1 0 0 2 0 1 1 0 3 0 0 1 0 3 ) , G 2 = 1 2 ( 1 0 0 a 0 0 0 0 0 0 0 0 a * 0 0 1 ) , G 3 = 1 4 ( 1 a 0 0 a * 1 0 0 0 0 1 b 0 0 b * 1 ) ,
G = 1 4 ( 1 0 μ S 0 0 1 0 μ I μ S 0 1 0 0 μ I 0 1 ) .
G 1 = 1 2 ( 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ) G s ( r ) = 1 2 ( 1 0 0 1 ) V 0 = 0 U ^ 12 G 2 = 1 2 ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ) G s ( r ) = 1 2 ( 1 0 0 1 ) V 0 = 0 U ^ 23 G 3 = 1 2 ( 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ) G s ( r ) = ( 1 0 0 0 ) V 0 = 1 U ^ 34 G 4 = 1 4 ( 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ) G s ( r ) = 1 2 ( 1 1 1 1 ) V 0 = 1 ,

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