Abstract

From the basic fact that the matrix that describes a lossless multilayer belongs to the group SU(1, 1), which is locally isomorphic to the (2+1)-dimensional Lorentz group SO(2, 1), we present a natural identification of the parameters of the multilayer with those of a Lorentz transformation. We show that the phase that appears when one is studying the reflection and transmission of light on a compound multilayer is simply the relativistic Thomas rotation. We propose a simple optical experiment to determine the angle of this rotation.

© 1999 Optical Society of America

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References

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  1. Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).
  2. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, London, 1984).
  3. J. Sánchez-Mondragón, K. B. Wolf, Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).
  4. K. B. Wolf, Lie Methods in Optics II, Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1989).
    [CrossRef]
  5. A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  6. K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. (New York) 172, 1–25 (1986).
    [CrossRef]
  7. J. F. Cariñena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
    [CrossRef]
  8. A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  9. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  10. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
    [CrossRef]
  11. J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
    [CrossRef]
  12. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [CrossRef]
  13. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  14. M. D. Reid, D. F. Walls, “Generation of squeezed states in degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
    [CrossRef] [PubMed]
  15. J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
    [CrossRef] [PubMed]
  16. D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
    [CrossRef] [PubMed]
  17. A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
    [CrossRef]
  18. A. C. Hirshfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986).
    [CrossRef]
  19. J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
    [CrossRef]
  20. J. M. Vigoureux, D. V. Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
    [CrossRef]
  21. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).
  22. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  23. A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warsaw, 1977).
  24. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).
  25. D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  26. J. J. Monzón, L. L. Sánchez-Soto, “Characterization of symmetric, absorbing 50–50 beam splitters,” Appl. Opt. 35, 106–112 (1996).
    [CrossRef]
  27. A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
    [CrossRef]
  28. A. A. Ungar, “Successive Lorentz transformations of the electromagnetic field,” Found. Phys. 21, 569–589 (1991).
    [CrossRef]
  29. A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991).
    [CrossRef]
  30. H. C. Corben, “Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession,” Am. J. Phys. 61, 551–553 (1993).
    [CrossRef]
  31. M. W. P. Strandberg, “Special relativity completed: the source of some 2s in the magnitude of physical phenomena,” Am. J. Phys. 54, 321–331 (1986).
    [CrossRef]
  32. J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [CrossRef]
  33. P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
    [CrossRef]
  34. K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 351–393.

1999 (2)

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
[CrossRef]

1998 (1)

J. M. Vigoureux, D. V. Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

1997 (1)

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

1996 (2)

J. J. Monzón, L. L. Sánchez-Soto, “Characterization of symmetric, absorbing 50–50 beam splitters,” Appl. Opt. 35, 106–112 (1996).
[CrossRef]

J. F. Cariñena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

1993 (3)

H. C. Corben, “Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession,” Am. J. Phys. 61, 551–553 (1993).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

1992 (1)

1991 (3)

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

A. A. Ungar, “Successive Lorentz transformations of the electromagnetic field,” Found. Phys. 21, 569–589 (1991).
[CrossRef]

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991).
[CrossRef]

1989 (2)

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[CrossRef] [PubMed]

1986 (4)

J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
[CrossRef] [PubMed]

A. C. Hirshfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986).
[CrossRef]

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. (New York) 172, 1–25 (1986).
[CrossRef]

M. W. P. Strandberg, “Special relativity completed: the source of some 2s in the magnitude of physical phenomena,” Am. J. Phys. 54, 321–331 (1986).
[CrossRef]

1985 (2)

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

M. D. Reid, D. F. Walls, “Generation of squeezed states in degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Aravind, P. K.

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Barut, A. O.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warsaw, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Ben-Menahem, A.

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

Cariñena, J. F.

J. F. Cariñena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Corben, H. C.

H. C. Corben, “Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession,” Am. J. Phys. 61, 551–553 (1993).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 351–393.

Dragt, A. J.

Grossel, Ph.

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, London, 1984).

Han, D.

D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[CrossRef] [PubMed]

Hardekopf, E. E.

D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[CrossRef] [PubMed]

Hirshfeld, A. C.

A. C. Hirshfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986).
[CrossRef]

Jackson, D. A.

D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Kim, Y. S.

D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[CrossRef] [PubMed]

Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).

Klauder, J. R.

J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
[CrossRef] [PubMed]

Labeke, D. V.

J. M. Vigoureux, D. V. Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

Lekner, J.

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).

McCall, S. L.

J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
[CrossRef] [PubMed]

Metzger, F.

A. C. Hirshfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986).
[CrossRef]

Monzón, J. J.

Nasarre, J.

J. F. Cariñena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

Noz, M. E.

Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).

Perelomov, A.

A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

Ra¸czka, R.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warsaw, 1977).

Reid, M. D.

M. D. Reid, D. F. Walls, “Generation of squeezed states in degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
[CrossRef] [PubMed]

Sánchez-Mondragón, J.

J. Sánchez-Mondragón, K. B. Wolf, Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).

Sánchez-Soto, L. L.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, London, 1984).

Strandberg, M. W. P.

M. W. P. Strandberg, “Special relativity completed: the source of some 2s in the magnitude of physical phenomena,” Am. J. Phys. 54, 321–331 (1986).
[CrossRef]

Ungar, A. A.

A. A. Ungar, “Successive Lorentz transformations of the electromagnetic field,” Found. Phys. 21, 569–589 (1991).
[CrossRef]

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991).
[CrossRef]

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

Vigoureux, J. M.

J. M. Vigoureux, D. V. Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

Walls, D. F.

M. D. Reid, D. F. Walls, “Generation of squeezed states in degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
[CrossRef] [PubMed]

Wigner, E. P.

E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).

Wolf, K. B.

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. (New York) 172, 1–25 (1986).
[CrossRef]

J. Sánchez-Mondragón, K. B. Wolf, Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).

K. B. Wolf, Lie Methods in Optics II, Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1989).
[CrossRef]

Yurke, B.

J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
[CrossRef] [PubMed]

Am. J. Phys. (7)

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

A. C. Hirshfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986).
[CrossRef]

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991).
[CrossRef]

H. C. Corben, “Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession,” Am. J. Phys. 61, 551–553 (1993).
[CrossRef]

M. W. P. Strandberg, “Special relativity completed: the source of some 2s in the magnitude of physical phenomena,” Am. J. Phys. 54, 321–331 (1986).
[CrossRef]

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

Ann. Phys. (New York) (1)

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. (New York) 172, 1–25 (1986).
[CrossRef]

Appl. Opt. (1)

Fortschr. Phys. (1)

J. F. Cariñena, J. Nasarre, “On the symplectic structures arising in geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
[CrossRef]

Found. Phys. (2)

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

A. A. Ungar, “Successive Lorentz transformations of the electromagnetic field,” Found. Phys. 21, 569–589 (1991).
[CrossRef]

J. Mod. Opt. (1)

J. M. Vigoureux, D. V. Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

J. M. Vigoureux, “The reflection of light by planar stratified media: the grupoid of amplitudes and a phase ‘Thomas precession,’ ” J. Phys. A 26, 385–393 (1993).
[CrossRef]

Opt. Commun. (1)

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

Phys. Rev. A (3)

M. D. Reid, D. F. Walls, “Generation of squeezed states in degenerate four-wave mixing,” Phys. Rev. A 31, 1622–1635 (1985).
[CrossRef] [PubMed]

J. R. Klauder, S. L. McCall, B. Yurke, “Squeezed states from nondegenerate four-wave mixing,” Phys. Rev. A 33, 3204–3209 (1986).
[CrossRef] [PubMed]

D. Han, E. E. Hardekopf, Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[CrossRef] [PubMed]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Other (11)

A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warsaw, 1977).

E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).

D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, London, 1984).

J. Sánchez-Mondragón, K. B. Wolf, Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).

K. B. Wolf, Lie Methods in Optics II, Vol. 352 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1989).
[CrossRef]

K. Creath, “Phase-measurement interferometric techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 351–393.

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Figures (2)

Fig. 1
Fig. 1

Wave vectors of the input [Ea(+) and Es(-)] and output [Ea(-) and Es(+)] fields in a multilayer sandwiched between two semi-infinite ambient (0) and substrate (m+1) media.

Fig. 2
Fig. 2

Composition of two lossless multilayers (a) M1 and (b) M2 and the corresponding boosts in SO(2, 1). (c) The compound multilayer M(12) induces a Thomas rotation of angle 2Ψ [clearly seen in SO(2, 1)], which can be measured from the transmission phase shift.

Equations (53)

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n0 sin θ0==nj sin θj==nm+1 sin θm+1.
Ea(-)=RasEa(+)+TsaEs(-),
Es(+)=TasEa(+)+RsaEs(-),
Ea(+)Ea(-)=1Tas 1-RsaRasX Es(+)Es(-),
X=TasTsa-RasRsa.
Mas=I01L1I12L2I(j-1)jLjLmIm(m+1),
Ras=|Ras|exp(iρ),Tas=|Tas|exp(iτ),
X=exp(i2τ),
Rsa=-Ras* exp(i2τ),
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*.
det Mas=nm+1 cos θm+1n0 cos θ0.
M=abb*a*,
xμ=Λνμxν
(x0)2-(x1)2-(x2)2
det Λ=±1.
X=x0x1-ix2x1+ix2x0.
X=MXM,
Λ(M)=|a|2+|b|22 Re(ab*)2 Im(ab*)2 Re(ab)Re(a2+b2)Im(a2-b2)-2 Im(ab)-Im(a2+b2)Re(a2-b2).
Λ1Λ2M(Λ1)M(Λ2)=M(Λ1Λ2).
Λ=L(β)R(ϕ)=γ-γβ cos θ-γβ sin θ-γβ cos θ1+(γ-1)cos2 θ(γ-1)cos θ sin θ-γβ sin θ(γ-1)cos θ sin θ1+(γ-1)sin2 θ 1000cos ϕ-sin ϕ0sin ϕcos ϕ,
γ=1/1-β2.
a=1Tas=(γ+1)/2 exp(-iϕ/2),
b*=RasTas=(γ-1)/2 exp[-i(ϕ/2-θ)].
γ=cosh ζ,β=tanh ζ,γβ=sinh ζ,
Ras=tanh(ζ/2)exp(iθ),
Tas=sech(ζ/2)exp(iϕ/2).
ρ=θ,τ=ϕ/2.
M(12)=M1M2=[1+R1*R2 exp(i2τ1)]/T1T2[R1*+R2* exp(-i2τ1)]/T1*T2*[R1+R2 exp(i2τ1)]/T1T2[1+R1R2* exp(-i2τ1)]/T1*T2*.
R(12)=R1+R2 exp(i2τ1)1+R1*R2 exp(i2τ1),
T(12)=T1T21+R1*R2 exp(i2τ1).
M(21)=M2M1=[1+R1R2* exp(i2τ2)]/T1T2[R1* exp(-i2τ2)+R2*]/T1*T2*(R1 exp(i2τ2)+R2]/T1T2[1+R1*R2 exp(-i2τ2)]/T1*T2*,
R(21)=R1 exp(i2τ2)+R21+R1R2* exp(i2τ2),
T(21)=T1T21+R1R2* exp(i2τ2).
ρ1-τ1=ρ2-τ2
τ1=-τ2
τ1=τ2=0.
R(12)=R1+R21+R1*R2,
R(21)=R1+R21+R1R2*.
R(21)=R(12) exp(-i2Ψ),
Ψ=arg(1+R1R2*).
T(21)=T(12)*=T(12) exp(-i2Ψ),
Ψ=arg T(12).
L1(β1)L2(β2)=L(β)R(nˆ, Φ);
nˆ=β2×β1|β2×β1|,tan(Φ/2)=sin ΘK+cos Θ,
K2=γ1+1γ1-1 γ2+1γ2-1=1tanh2(ζ1/2)tanh2(ζ2/2),
β=1-1/Γ2,
Γ=γ1γ2(1+β1β2)=γ1γ2(1+β1β2 cos Θ),
M=HU,
M=HU=1/|T|R*/|T|R/|T|1/|T| exp(-iτ)00exp(iτ),
H1H2=HU=|1+R1*R2|/T1T2(R1*+R2*)exp(-iΨ)/T1T2(R1+R2)exp(iΨ)/T1T2|1+R1*R2|/T1T2 exp(-iΨ)00exp(iΨ).
2Ψ=2 arg(1+R1R2*)=2 arg T(12).
|R(12)|=tanh(ζ/2)=[(Γ-1)/(Γ+1)]1/2,
arg R(12)=arg(R1+R2)+Ψ.

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