Abstract

A measure for the twist of Gaussian light is expressed in terms of the second-order moments of the Wigner distribution function. The propagation law for these second-order moments between the input plane and the output plane of a first-order optical system is used to express the twist in one plane in terms of moments in the other plane. Although in general the twist in one plane is determined not only by the twist in the other plane but also by other combinations of the moments, several special cases exist for which a direct relationship between the twists can be formulated. Three such cases, for which zero twist is preserved, are considered: (i) propagation between conjugate planes, (ii) adaptation of the signal to the system, and (iii) the case of symplectic Gaussian light.

© 2000 Optical Society of America

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References

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  1. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  2. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]
  3. K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  4. A. T. Friberg, B. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  5. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  6. R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
    [CrossRef]
  7. R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
    [CrossRef]
  8. M. J. Bastiaans, “On the propagation of the twist of Gaussian light in first-order optical systems,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 121–125.
  9. M. J. Bastiaans, “Zero twist of Gaussian light in first-order optical systems,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 112–113 (1999).
    [CrossRef]
  10. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  11. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  12. M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
    [CrossRef]
  13. M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.
  14. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
    [CrossRef]
  15. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  16. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  17. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  18. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  19. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  20. M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
    [CrossRef]
  21. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
    [CrossRef] [PubMed]
  22. R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
    [CrossRef] [PubMed]
  23. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  24. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  25. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  26. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  27. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).
  28. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

1998 (1)

1996 (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

1994 (3)

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

A. T. Friberg, B. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

1993 (3)

1992 (1)

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

1991 (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

1988 (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1987 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (1)

1977 (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

1976 (1)

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1968 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

M. J. Bastiaans, “On the propagation of the twist of Gaussian light in first-order optical systems,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 121–125.

M. J. Bastiaans, “Zero twist of Gaussian light in first-order optical systems,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 112–113 (1999).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Dutta, B.

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

Friberg, A. T.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

A. T. Friberg, B. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

Mandel, L.

Mukunda, N.

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sundar, K.

Tervonen, B.

Turunen, J.

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

Opt. Commun. (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Quantum Electron. (1)

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

Optik (Stuttgart) (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (3)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems—U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

Other (6)

M. J. Bastiaans, “On the propagation of the twist of Gaussian light in first-order optical systems,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), pp. 121–125.

M. J. Bastiaans, “Zero twist of Gaussian light in first-order optical systems,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 112–113 (1999).
[CrossRef]

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 65–87.

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Equations (46)

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Γ(r1, r2)=1πdet G1 exp-14r1+r2r1-r2tG1-iH-iHtG2×r1+r2r1-r2,
Γ(r1, r2)=1πdet G1×exp[-14(r1-r2)t(G2-G1)(r1-r2)]×exp{-12r1t[G1-i12(H+Ht)]r1}×exp{-12r2t[G1+i12(H+Ht)]r2}×exp[-12r1ti(H-Ht)r2].
F(r, q)=-Γ(r+12r, r-12r)exp(-iqtr)dr,
F(r, q)=4det G1det G2 exp-rqt×G1+HG2-1Ht-HG2-1-G2-1HtG2-1rq.
M--F(r, q)drdq2π=--rrtrqtqrtqqtF(r, q)drdq2π.
M=RPPtQ=12G1+HG2-1Ht-HG2-1-G2-1HtG2-1-1=12G1-1G1-1HHtG1-1G2+HtG1-1H,
R=12G1-1,P=12G1-1H,
Q=12(G2+HtG1-1H).
2qrtMqr=rtG2r+(q+Hr)tG1-1(q+Hr)
G1=12R-1,G2=2(Q-PtR-1P),H=R-1P.
PR-RPt=R(H-Ht)R=14G1-1(H-Ht)G1-1=14(H-Ht)det G1-1.
Fo(r, q)=Fi(Ar+Bq, Cr+Dq),
riqi=Troqo=ABCDroqo.
ABt=BAt,CtA=AtC,ADt-BCt=I
BtD=DtB,DCt=CDt,AtD-CtB=I.
Γo(r1o, r2o)=--h(r1o, r1i)Γi(r1i, r2i)h*(r2o, r2i)×dr1idr2i,
expi12roritLooLoiLioLiirori,
qo-qi=Lrori=LooLoiLioLiirori.
Loo=Loot=-B-1A,Loi=Liot=B-1,
Lii=Liit=-DB-1,
Mi=TMoTt.
RiPiPitQi=ABCDRoPoPotQoAtCtBtDt,
Ri=ARoAt+APoBt+BPotAt+BQoBt,
Pi=ARoCt+APoDt+BPotCt+BQoDt,
Qi=CRoCt+CPoDt+DPotCt+DQoDt.
G1i-1(Hi-Hit)G1i-1=(A+BHot)G1o-1(Ho-Hot)G1o-1(A+BHot)t-(A+BHot)G1o-1G2oBt+BG2oG1o-1(A+BHot)t.
G1o-1(Ho-Hot)G1o-1=(D-HiB)tG1i-1(Hi-Hit)G1i-1(D-HiB)+(D-HiB)tG1i-1G2iB-BtG2iG1i-1(D-HiB).
(A+BHot)G1o-1G2oBt=BG2oG1o-1(A+BHot)t,
(D-HiB)tG1i-1G2iB=BtG2iG1i-1(D-HiB);
G1o=AtG1iA,G2o=AtG2iA,
Ho=AtHiA-CtA.
A+BHot=0,
D-HiB=0,
Ri=BQoBt-ARoAt,
Pi=BQoDt+ARoHoDt.
PiB=RiD.
G2o-1=BtG1iB,G1o-1=BtG2iB.
exp[i12(r1otHor1o-r2otHor2o)]
×--exp[i(r1otLoir1i-r2otLoir2i)]×exp[-12r1itG1ir1i-14(r1i-r2i)t(G2i-G1i)×(r1i-r2i)-12r2itG1ir2i]dr1idr2i,
IB0I=I0B-1I0B-B-10I0B-1I.
Hi±iGi=[C+D(Ho±iGo)][A+B(Ho±iGo)]-1
Ho±iGo=-[D-(Hi±iGi)B]-1[C-(Hi±iGi)A]
PiRi-RiPit=ARo(CtA-AtC)RoAt+ARo(CtA-AtC)PoBt+ARo(CtB-AtD)PotAt+ARo(CtB-AtD)QoBt+APo(DtA-BtC)RoAt+APo(DtA-BtC)PoBt+APo(DtB-BtD)PotAt+APo(DtB-BtD)QoBt+BPot(CtA-AtC)RoAt+BPot(CtA-AtC)PoBt+BPot(CtB-AtD)PotAt+BPot(CtB-AtD)QoBt+BQo(DtA-BtC)RoAt+BQo(DtA-BtC)PoBt+BQo(DtB-BtD)PotAt+BQo(DtB-BtD)QoBt.
PiRi-RiPit=(APoRoAt-ARoPotAt)+(APoPoBt-BPotPotAt)+(BQoRoAt-ARoQoBt)+(BQoPoBt-BPotQoBt).
4(PiRi-RiPit)=AG1-1(H-Ht)G1-1At+(AG1-1HG1-1HBt-BHtG1-1HtG1-1At)+(BG2G1-1At-AG1-1G2Bt)+(BHtG1-1HG1-1At-AG1-1HtG1-1HBt)+(BG2G1-1HBt-BHtG1-1Bt)+BHtG1-1(H-Ht)G1-1HBt.
4(PiRi-RiPit)=(A+BHt)G1-1(H-Ht)G1-1(A+BHt)t-(A+BHt)G1-1G2Bt+BG2G1-1(A+BHt)t.

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