Abstract

We suggest a method to access the second-order, or two-point, Stokes parameters of a partially coherent and partially polarized Gaussian model optical field from an intensity interferometry experiment. Through a remarkably simple experimental arrangement, it is possible to measure the two-point and one-point Stokes parameters simultaneously, allowing the reconstruction of the coherence matrix and the polarization matrix, thus completely characterizing the optical field both statistically and locally on the observation plane. Developments, automation, and applications are pointed out.

© 2014 Optical Society of America

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References

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  1. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, Vol. 1 (J. Johnson, 1807), p. 892.
  2. A. J. Fresnel, Annales de Chimieet de Physique 2, 239 (1816).
  3. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  4. J. Ellis and A. Dogariu, Opt. Lett. 29, 536 (2004).
    [CrossRef]
  5. O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
    [CrossRef]
  6. J. Tervo, T. Setala, A. Roueff, P. Refregier, and A. T. Friberg, Opt. Lett. 34, 3074 (2009).
    [CrossRef]
  7. O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).
  8. O. V. Angelsky, S. G. Hanson, C. Y. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, Opt. Express 17, 15623 (2009).
    [CrossRef]
  9. T. Setala, J. Tervo, and A. T. Friberg, Opt. Lett. 31, 2669 (2006).
    [CrossRef]
  10. B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Lett. 34, 719 (2009).
    [CrossRef]
  11. H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
    [CrossRef]
  12. B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Commun. 282, 3059 (2009).
    [CrossRef]
  13. B. Kanseri and H. C. Kandpal, Opt. Lett. 33, 2410 (2008).
    [CrossRef]
  14. B. Kanseri and H. C. Kandpal, Opt. Commun. 283, 4558 (2010).
    [CrossRef]
  15. O. V. Angelsky and P. V. Polyanskii, Appl. Opt. 51, C100 (2012).
    [CrossRef]
  16. E. Wolf, ed., in Progress in Optics (Elsevier, 2007), Vol. 50, pp. 251–273.
  17. J. Tervo, T. Setala, and A. T. Friberg, Opt. Lett. 37, 151 (2012).
    [CrossRef]
  18. R. Brown and R. Twiss, Nature 177, 27 (1956).
    [CrossRef]
  19. R. Brown and R. Twiss, Proc. R. Soc. London 242, 300 (1957).
    [CrossRef]
  20. R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
    [CrossRef]
  21. R. Brown and R. Twiss, Nature 178, 1046 (1956).
    [CrossRef]
  22. R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
    [CrossRef]
  23. W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
    [CrossRef]
  24. J. Howard, Plasma Phys. Controlled Fusion 48, 777 (2006).
    [CrossRef]
  25. J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
    [CrossRef]
  26. T. Setala, J. Tervo, and A. Friberg, Opt. Lett. 29, 328 (2004).
    [CrossRef]
  27. T. Hassinen, J. Tervo, T. Setala, and A. T. Friberg, Opt. Express 19, 15188 (2011).
    [CrossRef]
  28. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
    [CrossRef]

2013 (1)

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

2012 (2)

2011 (2)

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

T. Hassinen, J. Tervo, T. Setala, and A. T. Friberg, Opt. Express 19, 15188 (2011).
[CrossRef]

2010 (1)

B. Kanseri and H. C. Kandpal, Opt. Commun. 283, 4558 (2010).
[CrossRef]

2009 (4)

2008 (2)

B. Kanseri and H. C. Kandpal, Opt. Lett. 33, 2410 (2008).
[CrossRef]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

2006 (2)

J. Howard, Plasma Phys. Controlled Fusion 48, 777 (2006).
[CrossRef]

T. Setala, J. Tervo, and A. T. Friberg, Opt. Lett. 31, 2669 (2006).
[CrossRef]

2005 (2)

O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
[CrossRef]

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

2004 (2)

2003 (2)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

1957 (2)

R. Brown and R. Twiss, Proc. R. Soc. London 242, 300 (1957).
[CrossRef]

R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
[CrossRef]

1956 (2)

R. Brown and R. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Brown and R. Twiss, Nature 177, 27 (1956).
[CrossRef]

1816 (1)

A. J. Fresnel, Annales de Chimieet de Physique 2, 239 (1816).

Angelsky, O. V.

O. V. Angelsky and P. V. Polyanskii, Appl. Opt. 51, C100 (2012).
[CrossRef]

O. V. Angelsky, S. G. Hanson, C. Y. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, Opt. Express 17, 15623 (2009).
[CrossRef]

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Bachin-skiy, V. T.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Baldwin, K. G. H.

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Bogatyryova, H. V.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Boichuk, T. M.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Brown, R.

R. Brown and R. Twiss, Proc. R. Soc. London 242, 300 (1957).
[CrossRef]

R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
[CrossRef]

R. Brown and R. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Brown and R. Twiss, Nature 177, 27 (1956).
[CrossRef]

Cramer, J.

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

Dall, R. G.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Dogariu, A.

Ellis, J.

Felde, C. V.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Fresnel, A. J.

A. J. Fresnel, Annales de Chimieet de Physique 2, 239 (1816).

Friberg, A.

Friberg, A. T.

Gorodyns’ka, N. V.

Gorsky, M. P.

Hanson, S. G.

Hassinen, T.

Hodgman, S. S.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Howard, J.

J. Howard, Plasma Phys. Controlled Fusion 48, 777 (2006).
[CrossRef]

James, D. F. V.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Johns-son, M. T.

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Kandpal, H. C.

B. Kanseri and H. C. Kandpal, Opt. Commun. 283, 4558 (2010).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Lett. 34, 719 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Commun. 282, 3059 (2009).
[CrossRef]

B. Kanseri and H. C. Kandpal, Opt. Lett. 33, 2410 (2008).
[CrossRef]

Kanseri, B.

B. Kanseri and H. C. Kandpal, Opt. Commun. 283, 4558 (2010).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Commun. 282, 3059 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Lett. 34, 719 (2009).
[CrossRef]

B. Kanseri and H. C. Kandpal, Opt. Lett. 33, 2410 (2008).
[CrossRef]

Kheruntsyan, K. V.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

Korotkova, O.

Lamberton, T.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

Little, A.

R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
[CrossRef]

Manning, A. G.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Miller, G.

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Polyanskii, P. V.

O. V. Angelsky and P. V. Polyanskii, Appl. Opt. 51, C100 (2012).
[CrossRef]

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Rath, S.

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Commun. 282, 3059 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Lett. 34, 719 (2009).
[CrossRef]

Refregier, P.

Roueff, A.

Roychowdhury, H.

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

RuGway, W.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

Setala, T.

Shirai, T.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Tervo, J.

Truscott, A. G.

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Twiss, R.

R. Brown and R. Twiss, Proc. R. Soc. London 242, 300 (1957).
[CrossRef]

R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
[CrossRef]

R. Brown and R. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Brown and R. Twiss, Nature 177, 27 (1956).
[CrossRef]

Ushenko, A. G.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Wolf, E.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
[CrossRef]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

Wu, J.

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

Yoon, J.

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

Young, T.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, Vol. 1 (J. Johnson, 1807), p. 892.

Zenkova, C. Y.

O. V. Angelsky, S. G. Hanson, C. Y. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, Opt. Express 17, 15623 (2009).
[CrossRef]

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

Annales de Chimieet de Physique (1)

A. J. Fresnel, Annales de Chimieet de Physique 2, 239 (1816).

Appl. Opt. (1)

J. Opt. A (1)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Nat. Commun. (1)

R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johns-son, K. G. H. Baldwin, and A. G. Truscott, Nat. Commun. 2, 291 (2011).
[CrossRef]

Nature (3)

R. Twiss, A. Little, and R. Brown, Nature 180, 324 (1957).
[CrossRef]

R. Brown and R. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Brown and R. Twiss, Nature 177, 27 (1956).
[CrossRef]

Opt. Commun. (3)

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, Opt. Commun. 282, 3059 (2009).
[CrossRef]

B. Kanseri and H. C. Kandpal, Opt. Commun. 283, 4558 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

J. Cramer, G. Miller, J. Wu, and J. Yoon, Phys. Rev. Lett. 94, 102302 (2005).
[CrossRef]

W. RuGway, A. G. Manning, S. S. Hodgman, R. G. Dall, A. G. Truscott, T. Lamberton, and K. V. Kheruntsyan, Phys. Rev. Lett. 111, 093601 (2013).
[CrossRef]

Plasma Phys. Controlled Fusion (1)

J. Howard, Plasma Phys. Controlled Fusion 48, 777 (2006).
[CrossRef]

Proc. R. Soc. London (1)

R. Brown and R. Twiss, Proc. R. Soc. London 242, 300 (1957).
[CrossRef]

Other (3)

E. Wolf, ed., in Progress in Optics (Elsevier, 2007), Vol. 50, pp. 251–273.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts, Vol. 1 (J. Johnson, 1807), p. 892.

O. V. Angelsky, P. V. Polyanskii, I. I. Mokhun, C. Y. Zenkova, H. V. Bogatyryova, C. V. Felde, V. T. Bachin-skiy, T. M. Boichuk, and A. G. Ushenko, Optical Measurements: Polarization and Coherence of Light Fields, Modern Metrology Concerns, D. L. Cocco, ed. (In-Tech, 2012).

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Figures (1)

Fig. 1.
Fig. 1.

Suggested setup, prior to HBT interferometer.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

με2(r⃗1,r⃗2,ω)=i,j|Wij(r⃗1,r⃗2,ω)|2I(r⃗1,ω)I(r⃗2,ω),
Wij(r⃗1,r⃗2,ω)=Ei*(r⃗1,ω)Ej(r⃗2,ω).
ΔI(r⃗1)ΔI(r⃗2)=με2(r⃗1,r⃗2)I(r⃗1)I(r⃗2),
Γ(r⃗1,r⃗2)=i,j|Ei*(r⃗1)Ej(r⃗2)|2.
E⃗res(r⃗)={[cos2θ+isin2θ]Ex(r⃗)+[(1i)cosθsinθ]Ey(r⃗)}x^,
Γ(r⃗1,r⃗2,θ)=|[cos4θ+sin4θ]Wxx(r⃗1,r⃗2)+[2cos2θsin2θ]Wyy(r⃗1,r⃗2)+[cos2θisin2θ][(1i)cosθsinθ]Wxy(r⃗1,r⃗2)+[cos2θ+isin2θ][(1+i)cosθsinθ]Wyx(r⃗1,r⃗2)|2
Γ(r⃗1,r⃗2,θ)=14|[1+cos22θ]Wxx(r⃗1,r⃗2)+[1cos22θ]Wyy(r⃗1,r⃗2)+[sin4θ2isin2θ]Wxy(r⃗1,r⃗2)+[sin4θ2+isin2θ]Wyx(r⃗1,r⃗2)|2.
S0(r⃗1,r⃗2)=Wxx(r⃗1,r⃗2)+Wyy(r⃗1,r⃗2)S1(r⃗1,r⃗2)=Wxx(r⃗1,r⃗2)Wyy(r⃗1,r⃗2)S2(r⃗1,r⃗2)=Wyx(r⃗1,r⃗2)+Wxy(r⃗1,r⃗2)S3(r⃗1,r⃗2)=i[Wyx(r⃗1,r⃗2)Wxy(r⃗1,r⃗2)],
Γ(r⃗1,r⃗2,θ)=14|S0(r⃗1,r⃗2)+S1(r⃗1,r⃗2)cos22θ+S2(r⃗1,r⃗2)sin4θ2+S3(r⃗1,r⃗2)sin2θ|2.
με2(r⃗1,r⃗2)=i=03|Si(r⃗1,r⃗2)|22S0(r⃗1,r⃗1)S0(r⃗2,r⃗2),
P(r⃗1,r⃗2)=i=13|Si(r⃗1,r⃗2)|2S0(r⃗1,r⃗2).
I(r⃗,θ)=12[S0(r⃗)+S1(r⃗)cos22θ+S2(r⃗)sin4θ2+S3(r⃗)sin2θ].
S0(r⃗1,r⃗2)=Γ(θ3)1/2+Γ(θ4)1/2S1(r⃗1,r⃗2)=2Γ(θ1)1/2Γ(θ3)1/2Γ(θ4)1/2S2(r⃗1,r⃗2)=2Γ(θ1)1/2+(1+2)Γ(θ3)1/2+(12)Γ(θ4)1/24Γ(θ2)1/2S3(r⃗1,r⃗2)=Γ(θ3)1/2Γ(θ4)1/2,
S0(r⃗)=I(θ3)+I(θ4)S1(r⃗)=2I(θ1)I(θ3)I(θ4)S2(r⃗)=2I(θ1)+(1+2)I(θ3)+(12)I(θ4)4I(θ2)S3(r⃗)=I(θ3)I(θ4).

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