Abstract

We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2, 1) in a (2+1)-dimensional space–time as well as the group SL(2, ℝ) underlying the structure of the ABCD law in geometrical optics. By resorting to the Iwasawa decomposition, we represent the action of any multilayer as the product of three matrices of simple interpretation. This group-theoretical structure allows us to introduce bilinear transformations in the complex plane. The concept of multilayer transfer function naturally emerges, and its corresponding properties in the unit disk are studied. We show that the Iwasawa decomposition is reflected at this geometrical level in three simple actions that can be considered the basic pieces for a deeper understanding of the multilayer behavior. We use the method to analyze in detail a simple practical example.

© 2002 Optical Society of America

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References

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  1. H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1986).
  2. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  3. J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).
  4. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [CrossRef]
  5. J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [CrossRef]
  6. J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
    [CrossRef]
  7. J. J. Monzón, L. L. Sánchez-Soto, “Multilayer optics as an analog computer for testing special relativity,” Phys. Lett. A 262, 18–26 (1999).
    [CrossRef]
  8. J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
    [CrossRef]
  9. H. S. M. Coxeter, Non-Euclidean Geometry (University of Toronto Press, Toronto, 1968).
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  11. D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
    [CrossRef]
  12. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  13. M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
    [CrossRef]
  14. J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, “Basic factorization for multilayers,” Opt. Lett. 26, 370–372 (2001).
    [CrossRef]
  15. When ambient (0) and substrate (m+1)media are different, the angles θ0and θm+1are conected by Snell law n0 sin θ0=nm+1 sin θm+1,where njdenotes the refractive index of the jthmedium.
  16. I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics XLI, E. Wolf, ed. (North-Holland, Amsterdam, 2000), pp. 181–282.
  17. H. H. Arsenault, B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983).
    [CrossRef]
  18. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef] [PubMed]
  19. J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [CrossRef]
  20. A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN, Warszaw, 1977).
  21. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978).
  22. H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [CrossRef]
  23. M. Nazarathy, J. Shamir, “First order systems—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  24. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  25. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  26. R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef] [PubMed]
  27. G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
    [CrossRef]
  28. R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
    [CrossRef]
  29. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  30. J. F. Cariñena, J. Nasarre, “On symplectic structures arising from geometric optics,” Fortschr. Phys. 44, 181–198 (1996).
    [CrossRef]
  31. V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
    [CrossRef]
  32. A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (MIR, Moscow, 1988), Sec. 1.4.

2001 (2)

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, “Basic factorization for multilayers,” Opt. Lett. 26, 370–372 (2001).
[CrossRef]

1999 (4)

J. J. Monzón, L. L. Sánchez-Soto, “Multilayer optics as an analog computer for testing special relativity,” Phys. Lett. A 262, 18–26 (1999).
[CrossRef]

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]

J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

1998 (2)

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

1996 (2)

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

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

1995 (1)

1994 (1)

1993 (2)

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

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]

1985 (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983 (1)

1982 (1)

1981 (1)

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

1947 (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

Abe, S.

Agarwal, G. S.

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

Arsenault, H. H.

Azzam, R. M. A.

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

Bacry, H.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bargmann, V.

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

Barut, A. O.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN, Warszaw, 1977).

Bashara, N. M.

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

Cadilhac, M.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Cariñena, J. F.

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

Cohen, N.

Coxeter, H. S. M.

H. S. M. Coxeter, Non-Euclidean Geometry (University of Toronto Press, Toronto, 1968).

Fomenko, A.

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (MIR, Moscow, 1988), Sec. 1.4.

Franta, D.

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics XLI, E. Wolf, ed. (North-Holland, Amsterdam, 2000), pp. 181–282.

Han, D.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Helgason, S.

S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978).

Kim, Y. S.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Kubota, J. H.

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

Lekner, J.

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

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1986).

Macukow, B.

Mischenko, A.

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (MIR, Moscow, 1988), Sec. 1.4.

Monzón, J. J.

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, “Basic factorization for multilayers,” Opt. Lett. 26, 370–372 (2001).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Multilayer optics as an analog computer for testing special relativity,” Phys. Lett. A 262, 18–26 (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]

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, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

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

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Nakazawa, M.

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

Nasarre, J.

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

Nazarathy, M.

Noz, M. E.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Ohli´dal, I.

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics XLI, E. Wolf, ed. (North-Holland, Amsterdam, 2000), pp. 181–282.

Ra¸czka, R.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN, Warszaw, 1977).

Sahara, A.

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

Sánchez-Soto, L. L.

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, “Basic factorization for multilayers,” Opt. Lett. 26, 370–372 (2001).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Multilayer optics as an analog computer for testing special relativity,” Phys. Lett. A 262, 18–26 (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]

J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
[CrossRef]

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

Shamir, J.

Sheridan, J. T.

Simon, R.

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Tamura, K.

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Yonte, T.

Ann. Math. (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Fortschr. Phys. (1)

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

IEEE J. Quantum Electron. (1)

M. Nakazawa, J. H. Kubota, A. Sahara, K. Tamura, “Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,” IEEE J. Quantum Electron. QE34, 1075–1081 (1998).
[CrossRef]

J. Mod. Opt. (1)

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Commun. (3)

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

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

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

Opt. Lett. (2)

Phys. Lett. A (2)

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representations of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Multilayer optics as an analog computer for testing special relativity,” Phys. Lett. A 262, 18–26 (1999).
[CrossRef]

Phys. Rev. A (2)

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

Other (10)

A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (MIR, Moscow, 1988), Sec. 1.4.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1986).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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

H. S. M. Coxeter, Non-Euclidean Geometry (University of Toronto Press, Toronto, 1968).

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

When ambient (0) and substrate (m+1)media are different, the angles θ0and θm+1are conected by Snell law n0 sin θ0=nm+1 sin θm+1,where njdenotes the refractive index of the jthmedium.

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics XLI, E. Wolf, ed. (North-Holland, Amsterdam, 2000), pp. 181–282.

A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN, Warszaw, 1977).

S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978).

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Figures (4)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Unit hyperboloids defined in Eq. (25), with K=1, which represent the space of field states for SO(2, 1). In each one we have plotted a typical orbit for the matrices: (a) ΛK, (b) ΛA, (c) ΛN. In all the figures we have performed stereographic projection from the south pole S of the hyperboloid to obtain the unit disk in the plane e0=0 and the corresponding orbits, which represent the actions of the SU(1, 1) matrices: (a) K, (b) A, and (c) N.

Fig. 3
Fig. 3

(a) Plot of several orbits in the unit disk of the elements of the Iwasawa decomposition K, A, and N for SU(1, 1) (from left to right, respectively). (b) Corresponding orbits in the upper complex semiplane for the Iwasawa decomposition K, A, and N for SL(2, ℝ).

Fig. 4
Fig. 4

Geometrical representation in the unit disk of the action of a single glass plate with the parameters indicated in the text. The point zs is transformed by the plate into the point za. We indicate the three orbits given by the Iwasawa decomposition and, with a thick line, the trajectory associated with the plate action.

Equations (52)

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

E=E(+)E(-),
Ea=MasEs.
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*αββ*α*,
Ras=|Ras|exp(iρ),Tas=|Tas|exp(iτ)
det Mas=|α|2-|β|2=1-|Ras|2|Tas|2=+1.
Mas=K(ϕ)A(ξ)N(ν),
K(ϕ)=exp(iϕ/2)00exp(-iϕ/2),
A(ξ)=cosh(ξ/2)i sinh(ξ/2)-i sinh(ξ/2)cosh(ξ/2),
N(ν)=1-iν/2ν/2ν/21+iν/2.
ϕ/2=arg(α+iβ),
ξ/2=ln(1/|α+iβ|),
ν/2=Re(αβ*)/|α+iβ|2,
|Ea(+)|2-|Ea(-)|2=|Es(+)|2-|Es(-)|2,
U=121ii1
Ea=MasEs,
E=E(+)E(-)=UE=12E(+)+iE(-)E(-)+iE(+),
Mas=U MasU-1=abcd,
a=Re(α)+Im(β),b=Im(α)+Re(β),
c=-Im(α)+Re(β),d=Re(α)-Im(β).
Mas(2)Mas(1)=UMas(2)U-1UMas(1)U-1=UMas(2)Mas(1)U-1.
Mas=K(ϕ)A(ξ)N(ν),
K(ϕ)=cos(ϕ/2)sin(ϕ/2)-sin(ϕ/2)cos(ϕ/2),
A(ξ)=exp(ξ/2)00exp(-ξ/2),
N(ν)=10ν1.
xμ=Λνμxν
x|y=x0y0-x1y1-x2y2
det Λ=±1.
X=xμσμ=x0x1-ix2x1+ix2x0,
X=MXM,
Λνμ(M)=12Tr(σμMσνM).
x0x1x2e0e1e2=(|E(+)|2+|E(-)|2)/2Re[E(+)*E(-)]Im[E(+)*E(-)].
(e0)2-(e1)2-(e2)2=K2.
Λ(Mas)=|α|2+|β|22 Re(αβ*)2 Im(αβ*)2 Re(αβ)Re(α2+β2)Im(α2-β2)-2 Im(αβ)-Im(α2+β2)Re(α2-β2).
ΛK(ϕ)=1000cos ϕsin ϕ0-sin ϕcos ϕ,
ΛA(ξ)=cosh ξ0-sinh ξ010-sinh ξ0cosh ξ,
ΛN(ν)=1+(ν2/2)ν-ν2/2ν1-νν2/2ν1-(ν2/2).
zs=Es(-)Es(+),za=Ea(-)Ea(+).
za=Φ[Mas,zs]=β*+α*zsα+βzs,
Φ[Mas, -α/β]=,Φ[Mas, ]=α*/β.
|za|2=|α|2|zs|2+|β|2+2 Re(βα*zs)|α|2+|β|2|zs|2+2 Re(βα*zs),
(|α|2-|β|2)(|zs|2-1)=|zs|2-1,
z=e1+ie21+e0=E(-)E(+),
z=Φ[K(ϕ), z]=zexp(-iϕ),
z=Φ[A(ξ), z]=z-i tanh(ξ/2)1+iz tanh(ξ/2),
z=Φ[N(ν), z]=z+(1+iz)ν/21+(z-i)ν/2.
w=Φ[U, z]=z+i1+iz,
w=Ψ[K(ϕ), w]=w-tan(ϕ/2)1+w tan(ϕ/2),
w=Ψ[A(ξ),w]=w exp(-ξ),
w=Ψ[N(ν), w]=w+ν.
Ras=r01[1-exp(-i2β1)]1-r012 exp(-i2β1),
Tas=(1-r012)exp(-iβ1)1-r012 exp(-i2β1),
β1=2πλn1d1 cos θ1.

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