Abstract

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that sorts the multilayers into three classes, each of whose properties are closely related to one (and only one) of the three basic matrices.

© 2001 Optical Society of America

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References

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  1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  2. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).
  3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  4. D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 219, 26 (1996).
    [CrossRef]
  5. J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
    [CrossRef]
  6. J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
    [CrossRef]
  7. J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2786 (1999).
    [CrossRef]
  8. J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Lett. 26, 370 (2001).
    [CrossRef]
  9. When ambient (0) and substrate (m+1) media are different, angles q0 and qm+1 are connected by Snell’s law, n0sinq 0=nm+1sin qm+1, where nj denotes the refractive index of the j th medium.
  10. I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41, p. 181.
    [CrossRef]
  11. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, New York, 1978).
  12. T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).
  13. Ref. 12 may also be found at arXiv.org e-Print archive, Preprint physics/0104050, http://xxx.lanl.gov.
  14. V. Bargmann, Ann. Math. 48, 568 (1947).
    [CrossRef]
  15. H. Bacry and M. Cadilhac, Phys. Rev. A 23, 2533 (1981).
    [CrossRef]
  16. R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
    [CrossRef]
  17. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 15, 2146 (1998).
    [CrossRef]
  18. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
    [CrossRef]

2001

2000

I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41, p. 181.
[CrossRef]

1999

1998

1996

D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 219, 26 (1996).
[CrossRef]

1988

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

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

1987

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

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

1984

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
[CrossRef]

1981

H. Bacry and M. Cadilhac, Phys. Rev. A 23, 2533 (1981).
[CrossRef]

1978

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

1947

V. Bargmann, Ann. Math. 48, 568 (1947).
[CrossRef]

Azzam, R. M. A.

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

Bacry, H.

H. Bacry and M. Cadilhac, Phys. Rev. A 23, 2533 (1981).
[CrossRef]

Bargmann, V.

V. Bargmann, Ann. Math. 48, 568 (1947).
[CrossRef]

Bashara, N. M.

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

Cadilhac, M.

H. Bacry and M. Cadilhac, Phys. Rev. A 23, 2533 (1981).
[CrossRef]

Cariñena, J. F.

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Franta, D.

I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41, p. 181.
[CrossRef]

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 219, 26 (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, and M. E. Noz, Phys. Lett. A 219, 26 (1996).
[CrossRef]

Lekner, J.

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

López-Lacasta, C.

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Monzón, J. J.

J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Lett. 26, 370 (2001).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2786 (1999).
[CrossRef]

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Mukunda, N.

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 15, 2146 (1998).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
[CrossRef]

Noz, M. E.

D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 219, 26 (1996).
[CrossRef]

Ohlídal, I.

I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41, p. 181.
[CrossRef]

Sánchez-Soto, L. L.

J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Lett. 26, 370 (2001).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2786 (1999).
[CrossRef]

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Simon, R.

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 15, 2146 (1998).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
[CrossRef]

Yeh, P.

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

Yonte, T.

J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Lett. 26, 370 (2001).
[CrossRef]

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Ann. Math.

V. Bargmann, Ann. Math. 48, 568 (1947).
[CrossRef]

J. Opt. Soc. Am. A

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2786 (1999).
[CrossRef]

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 15, 2146 (1998).
[CrossRef]

T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A (to be published).

Opt. Commun.

J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Opt. Lett.

Phys. Lett. A

D. Han, Y. S. Kim, and M. E. Noz, Phys. Lett. A 219, 26 (1996).
[CrossRef]

Phys. Rev. A

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 29, 3273 (1984).
[CrossRef]

H. Bacry and M. Cadilhac, Phys. Rev. A 23, 2533 (1981).
[CrossRef]

Other

When ambient (0) and substrate (m+1) media are different, angles q0 and qm+1 are connected by Snell’s law, n0sinq 0=nm+1sin qm+1, where nj denotes the refractive index of the j th medium.

I. Ohlídal and D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41, p. 181.
[CrossRef]

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

Ref. 12 may also be found at arXiv.org e-Print archive, Preprint physics/0104050, http://xxx.lanl.gov.

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

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

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

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Figures (1)

Fig. 1
Fig. 1

Plots of several orbits in the unit disk of the elements of the Iwasawa decomposition Kϕ, Aξ, and Nν for the group of multilayer transfer matrices.

Equations (17)

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

E=E+E-,
Ea=MasEs,
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*αββ*α*,
Mas=KϕAξNν,
Kϕ=expiϕ/200exp-iϕ/2,Aξ=coshξ/2isinhξ/2-isinhξ/2coshξ/2,Nν=1-iν/2ν/2ν/21+iν/2.
z=E-E+
za=ΦMas,zs=β*+α*zsα+βzs.
z=ΦMas,z,
z=-i Im α±Re α2-11/2β.
M^as=CMasC-1,
ΦCMasC-1,0=ΦCMas,zf=ΦC,zf=0.
U=121ii1.
Ea=MasEs,
E=E+E-=UE^+E^-=UCE+E-,
Mas=UM^asU-1=UCMasC-1U-1.
Mas=KϕAξNν,
Kϕ=cosϕ/2sinϕ/2-sinϕ/2cosϕ/2,Aξ=expξ/200exp-ξ/2,Nν=10ν1.

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