Abstract

A dissipative beam-splitter (BS) has been analyzed by modeling the losses in the BS due to the excitation of optical phonons. The losses are obtained in terms of the BS medium properties. The model simplifies the picture by treating the loss mechanism as a perturbation on the photon modes in a linear, non-lossy medium in the limit of small losses, instead of using the full field quantization in lossy, dispersive media. The model uses second order perturbation in the Markoff approximation and yields the Beer’s law for absorption in the first approximation, thus providing a microscopic description of the absorption coefficient. It is shown that the fluctuations in the modes get increased because of the losses. We show the existence of quantum interferences due to phase correlations between the input beams and it is shown that these correlations can result in loss quenching. Hence in spite of having such a dissipative medium, it is possible to design a lossless 50–50 BS at normal incidence which may have potential applications in laser optics and dielectric-coated mirrors.

© 1998 Optical Society of America

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References

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  1. For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”, Quantum. Opt. 1, 151 (1989).
    [Crossref]
  2. Also see the special issue of J. Mod. Opt.34 (1987).
  3. Also see the special issue of J. Opt. Soc. Am. B4 (1987).
  4. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
    [Crossref]
  5. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
    [Crossref] [PubMed]
  6. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
    [Crossref] [PubMed]
  7. B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A 38, 204 (1988).
    [Crossref] [PubMed]
  8. J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
    [Crossref]
  9. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
    [Crossref] [PubMed]
  10. M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
    [Crossref]
  11. J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
    [Crossref] [PubMed]
  12. U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A 48, 3265 (1993).
    [Crossref] [PubMed]
  13. S.-T. Ho and P. Kumar, “Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations”, J. Opt. Soc. Am. B 10, 1620 (1993).
    [Crossref]
  14. B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.  18, 487 (1992).
    [Crossref]
  15. B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306 (1992).
    [Crossref] [PubMed]
  16. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).
    [Crossref] [PubMed]
  17. U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod. Opt. 42, 1165 (1995).
  18. R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A 52, 4823 (1995).
    [Crossref] [PubMed]
  19. T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).
    [Crossref] [PubMed]
  20. Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
    [Crossref]
  21. N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .
  22. P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg, 1983).
  23. W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

1997 (1)

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

1996 (1)

T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).
[Crossref] [PubMed]

1995 (2)

U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod. Opt. 42, 1165 (1995).

R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A 52, 4823 (1995).
[Crossref] [PubMed]

1993 (3)

J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
[Crossref] [PubMed]

U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A 48, 3265 (1993).
[Crossref] [PubMed]

S.-T. Ho and P. Kumar, “Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations”, J. Opt. Soc. Am. B 10, 1620 (1993).
[Crossref]

1992 (2)

B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.  18, 487 (1992).
[Crossref]

B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306 (1992).
[Crossref] [PubMed]

1989 (2)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
[Crossref] [PubMed]

For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”, Quantum. Opt. 1, 151 (1989).
[Crossref]

1988 (2)

B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A 38, 204 (1988).
[Crossref] [PubMed]

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

1987 (2)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
[Crossref] [PubMed]

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
[Crossref]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
[Crossref] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).
[Crossref] [PubMed]

1966 (1)

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Aharanov, Y.

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Anhut, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .

Barnett, S. M.

B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.  18, 487 (1992).
[Crossref]

B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306 (1992).
[Crossref] [PubMed]

Ben-Aryeh, Y.

B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A 38, 204 (1988).
[Crossref] [PubMed]

Brendel, J.

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

Bruesch, P.

P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg, 1983).

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
[Crossref] [PubMed]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).
[Crossref] [PubMed]

Dakna, M.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

Falkoff, D.

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).
[Crossref] [PubMed]

Gruner, T.

T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).
[Crossref] [PubMed]

Ho, S.-T.

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
[Crossref] [PubMed]

Huttner, B.

B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306 (1992).
[Crossref] [PubMed]

B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.  18, 487 (1992).
[Crossref]

B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A 38, 204 (1988).
[Crossref] [PubMed]

Imoto, N.

J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
[Crossref] [PubMed]

Jeffers, J. R.

J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
[Crossref] [PubMed]

Knoll, L.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

Kumar, P.

Lange, R.

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod. Opt. 42, 1165 (1995).

U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A 48, 3265 (1993).
[Crossref] [PubMed]

Lerner, E.

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Loudon, R.

R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A 52, 4823 (1995).
[Crossref] [PubMed]

J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
[Crossref] [PubMed]

Louisell, W. H.

W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
[Crossref] [PubMed]

Martienssen, W.

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
[Crossref]

Matloob, R.

R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A 52, 4823 (1995).
[Crossref] [PubMed]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
[Crossref] [PubMed]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .

Opatrny, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
[Crossref] [PubMed]

Pendleton, H.

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Prasad, S.

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
[Crossref]

Saleh, B. A. E.

For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”, Quantum. Opt. 1, 151 (1989).
[Crossref]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
[Crossref] [PubMed]

Schutrumpf, S.

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

Scully, M. O.

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
[Crossref]

Teich, M. C.

For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”, Quantum. Opt. 1, 151 (1989).
[Crossref]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
[Crossref] [PubMed]

Welsch, D.-G.

T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).
[Crossref] [PubMed]

Welsh, D.-G.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

Yurke, B.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
[Crossref] [PubMed]

Ann. Phys. (1)

Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. 39, 498 (1966).
[Crossref]

Europhys. Lett (1)

B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.  18, 487 (1992).
[Crossref]

Europhys. Lett. (1)

J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett.,  5, 223 (1988).
[Crossref]

J. Mod. Opt. (1)

U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod. Opt. 42, 1165 (1995).

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun.,  62, 139 (1987).
[Crossref]

Phys. Rev. A (10)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A 33, 4033 (1986).
[Crossref] [PubMed]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantumum mechanical lossless beam splitter : SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).
[Crossref] [PubMed]

B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A 38, 204 (1988).
[Crossref] [PubMed]

B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306 (1992).
[Crossref] [PubMed]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).
[Crossref] [PubMed]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A 55, 3184 (1997).
[Crossref]

J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A 47, 3346 (1993).
[Crossref] [PubMed]

U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A 48, 3265 (1993).
[Crossref] [PubMed]

R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A 52, 4823 (1995).
[Crossref] [PubMed]

T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett.,  59, 2044 (1987).
[Crossref] [PubMed]

Quantum. Opt. (1)

For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”, Quantum. Opt. 1, 151 (1989).
[Crossref]

Other (5)

Also see the special issue of J. Mod. Opt.34 (1987).

Also see the special issue of J. Opt. Soc. Am. B4 (1987).

N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .

P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg, 1983).

W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

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Figures (2)

Figure 1.
Figure 1.

A beam-splitter with input and output light beams

Figure 2.
Figure 2.

Lattice displacements due to optical phonons : (a) transverse, (b) longitudinal.

Equations (45)

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

E 1 ( + ) = α 11 E 1 ( + ) + α 12 E 2 ( + )
E 2 ( + ) = α 21 E 1 ( + ) α 22 E 2 ( + )
a 1 a 2 = α 11 α 12 α 21 α 22 a 1 a 2
α 11 2 + α 21 2 = 1
α 12 2 + α 22 2 = 1
α 11 α 12 * + α 21 α 22 * = 0
a 1 a 2 = a 1 a 2 = α 11 α 12 α 21 α 22 a 1 a 2
= exp [ i ( ξ a 1 a 2 + η a 2 a 1 ) ]
α 11 α 12 α 21 α 22 = cos ηξ i ξ η sin ηξ i η ξ sin ηξ cos ηξ
1 , 0 = α 11 1,0 + α 21 0,1
n 1 , n 2 = 1 n 1 ! n 2 ! ( α 11 a 1 + α 21 a 2 ) n 1 ( α 12 a 1 + α 22 a 2 ) n 2 0,0
n 1 , n 2 ( Δ n 1 ) 2 n 1 , n 2 = α 11 2 α 12 2 ( n 1 + n 2 + 2 n 1 n 2 )
α , β ( Δ n 1 ) 2 α , β = ( α 11 2 + α 12 2 ) α 11 α + α 12 β 2
b k = 1 N k e i k · R [ M ω k 2 ħ u ( R ) + i 1 2 ħM ω k P ( R ) ] e ̂ k
b k = 1 N k e i k · R [ M ω k 2 ħ u ( R ) i 1 2 ħM ω k P ( R ) ] e ̂ k
R = l ħ ω l ( b l b l + 1 2 )
V i = q M A ( R ) · p
ψ = | j j f j ( r )
A ( R ) = k , λ ħ 2 ω V int [ e i k · R ̂ a + e i k . R ̂ a ] e ̂ λ
p ( R ) = l i ħ ω l M 2 N ( b l e i ω l t b l e i ω l t ) e ̂ l
V I = i = 1,2 l γ [ a i b l e i ( ω + ω l ) t a i b l e i ( ω ω l ) t a i b l e i ( ω ω l ) t + a i b l e i ( ω + ω l ) t ] e ̂ λ i · e ̂ l
T = o + R + V I
b l R = b l R = 0
b l b k R = b l b k R = 0
b l b k R = n ¯ ph ( ω l ) δ lk
b l b k R = [ n ¯ ph ( ω l ) + 1 ] δ lk
s = s ( 1 / 2 ) i , j ( a i a j s a i s a j a j s a i + s a j a i ) S ij
+ ( a i a j s a i s a j a j s a i + s a j a i ) S ij *
+ ( a i a j s a j s a i ) L ij ( a i s a j s a j a i ) L ij *
L ij = γ ( ω ) 2 cos θ il cos θ jl g ( ω )
S ij = n ¯ ph ( ω ) L ij
n 1 , n 2 a 1 a 1 n 1 , n 2 = n 1 ( 1 L 11 ) + S 11
n 1 , n 2 a 2 a 2 n 1 , n 2 = n 2 ( 1 L 22 ) + S 22
a i L = j ( δ ij L ij 2 ) a j NL
n 1 , n 2 Δ n 1 2 n 1 , n 2 = L 11 ( 4 L 11 ) n 2 + [ 2 S 11 + L 11 ( 2 S 11 1 ) ] n + S 11 ( 1 S 11 )
Δ I = I 0 [ 1 exp ( αl ) ] I 0 [ 1 ( 1 αl ) ] = I 0 αl
δ n 1 = L 11 n 1 = [ π 2 12 q 2 ρg ( ω ) ∊M V x ] A n 1 l
n 1 , n 2 a 1 a 1 n 1 , n 2 = ( α 11 2 n 1 + α 12 2 n 2 ) ( 1 L 11 )
Re [ L 12 ( α 11 | * α 21 n 1 + α 12 * α 22 n 2 ) ] + S 11
= L 2 [ ( n 1 + n 2 ) sin ξ sin θ sin 2 δ ξ ( n 1 n 2 ) ]
α , β a 1 a 1 α , β = α 11 α + α 12 β 2 ( 1 L 11 )
Re [ L 12 ( α 11 * α * + α 12 * β * ) ( α 21 α + α 22 β ) ]
= L 2 [ α 2 + β 2 + 2 αβ sin θ [ cos ( δ β δ α ) ( cos 2 ξ sin 2 ξ cos 2 δ ξ )
+ sin ( δ β δ α ) sin 2 ξ sin 2 δ ξ ]
= L 2 [ α 11 α + α 12 β 2 + α 21 α + α 22 β 2 + 2 sin θ Re { ( α 11 * α * + α 12 * β ) ( α 21 α + α 22 β ) } ]

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