Abstract

We re-elaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to separate multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics.

© 2003 Optical Society of America

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References

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  1. B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1997).
  2. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [CrossRef]
  3. J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [CrossRef]
  4. H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969).
  5. T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A 19, 603–609 (2002).
    [CrossRef]
  6. J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, J. F. Cariñena, “Geometrical setting for the classification of multilayers,” J. Opt. Soc. Am. A 19, 985–991 (2002).
    [CrossRef]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  8. 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]
  9. H. Kogelnik, “Imaging of optical modes–resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  10. 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]
  11. R. Melter, A. Rosenfeld, P. Bhattacharya, Vision Geometry (American Mathematical Society, Providence, R.I., 1991).
  12. K. A. Dunn, “Poincaré group as reflections in straight lines,” Am. J. Phys. 49, 52–55 (1981).
    [CrossRef]
  13. When ambient (0) and substrate (m+1)media are different, the angles θ0and θm+1are conected by Snell’s law n0sin θ0=nm+1sin θm+1,where njdenotes the refractive index of the jth medium.
  14. 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]
  15. 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]
  16. D. Pedoe, A Course of Geometry (Cambridge U. Press, Cambridge, UK, 1970).
  17. A. Mischenko, A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 1988), Sect. 1.4.
  18. L. L. Sánchez-Soto, J. J. Monzón, T. Yonte, J. F. Cariñena, “Simple trace criterion for classification of multilayers,” Opt. Lett. 26, 1400–1402 (2001).
    [CrossRef]
  19. J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, “Basic factorization for multilayers,” Opt. Lett. 26, 370–372 (2001).
    [CrossRef]
  20. B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

2002 (2)

2001 (3)

1999 (3)

1998 (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]

1996 (1)

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]

1981 (1)

K. A. Dunn, “Poincaré group as reflections in straight lines,” Am. J. Phys. 49, 52–55 (1981).
[CrossRef]

1965 (1)

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

Azzam, R. M. A.

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

Bashara, N. M.

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

Bhattacharya, P.

R. Melter, A. Rosenfeld, P. Bhattacharya, Vision Geometry (American Mathematical Society, Providence, R.I., 1991).

Cariñena, J. F.

Coxeter, H. S. M.

H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969).

Dunn, K. A.

K. A. Dunn, “Poincaré group as reflections in straight lines,” Am. J. Phys. 49, 52–55 (1981).
[CrossRef]

Fomenko, A.

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

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]

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]

López-Lacasta, C.

Melter, R.

R. Melter, A. Rosenfeld, P. Bhattacharya, Vision Geometry (American Mathematical Society, Providence, R.I., 1991).

Mischenko, A.

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

Monzón, J. J.

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]

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]

Pedoe, D.

D. Pedoe, A Course of Geometry (Cambridge U. Press, Cambridge, UK, 1970).

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Rosenfeld, A.

R. Melter, A. Rosenfeld, P. Bhattacharya, Vision Geometry (American Mathematical Society, Providence, R.I., 1991).

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.

Schutz, B. F.

B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1997).

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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]

Yonte, T.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Am. J. Phys. (1)

K. A. Dunn, “Poincaré group as reflections in straight lines,” Am. J. Phys. 49, 52–55 (1981).
[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]

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. A (4)

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]

Opt. Lett. (2)

Phys. Lett. A (1)

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]

Other (8)

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

R. Melter, A. Rosenfeld, P. Bhattacharya, Vision Geometry (American Mathematical Society, Providence, R.I., 1991).

H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969).

When ambient (0) and substrate (m+1)media are different, the angles θ0and θm+1are conected by Snell’s law n0sin θ0=nm+1sin θm+1,where njdenotes the refractive index of the jth medium.

B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1997).

D. Pedoe, A Course of Geometry (Cambridge U. Press, Cambridge, UK, 1970).

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

B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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Figures (7)

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

Outline of the unit hyperboloid and a geodesic on it. We also show how a hyperbolic line is obtained in the unit disk by stereographic projection, taking the south pole as projection center.

Fig. 3
Fig. 3

Plot of typical orbits in the unit disk for (a) canonical transfer matrices as given in Eq. (13), and (b) arbitrary transfer matrices.

Fig. 4
Fig. 4

Plot of the values of [Tr(Mas)]2 for a symmetric system made up of two identical plates (n1=1.7, d1=1 mm, θ0=π/4, λ = 0.6888 μm, and s-polarized light) separated by a spacer of variable phase thickness δ2.

Fig. 5
Fig. 5

Decomposition of the multilayer action in terms of two reflections in two intersecting lines for the same multilayer as in Fig. 4 with δ2=3 rad (elliptic case).

Fig. 6
Fig. 6

Decomposition of the multilayer action in terms of two reflections in two ultraparallel lines for the same multilayer as in Fig. 4 with δ2=1 rad (hyperbolic case).

Fig. 7
Fig. 7

Decomposition of the multilayer action in terms of two reflections in two parallel lines for the same multilayer as in Fig. 4 with δ2=0.4328 rad (parabolic case).

Equations (28)

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E=E(+)E(-)
Ea=MasEs.
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*αββ*α*,
e0=12 [|E(+)|2+|E(-)|2],
e1=Re[E(+)*E(-)],
e2=Im[E(+)*E(-)],
e3=12 [|E(+)|2-|E(-)|2],
(e0)2-(e1)2-(e2)2=(e3)2=const.
z=e1+ie21+e0=E(-)E(+).
za=Φ[Mas, zs]=β*+α*zsα+βzs,
(zA, zB|zC, zD)=(zA-zC)/(zB-zC)(zA-zD)/(zB-zD),
dH(z, z)=12 |ln(E, E|z, z)|.
zf=Φ[Mas, zf],
zf=12β {-2i Im(α)±[Tr(Mas)]2-4}.
Mˆas=CMasC-1,
Kˆ(φ)=exp(iφ/2)00exp(-iφ/2),
Aˆ(χ)=cosh(χ/2)i sinh(χ/2)-i sinh(χ/2)cosh(χ/2),
Nˆ(η)=1-iη/2η/2η/21+iη/2,
|z-w||z-w|=R2;
z=w+R2z*-w*=R2+wz*-w*wz*-w*.
z=wz*-1z*-w*.
za=Φ*[Mas, zs]=β*+α*zs*α+βzs*.
z=Φ*[Iw, z],
Iw=-iw*/Ri/R-i/Riw/R.
z=Φ[IwIw*, z].
z  1z*,
(1/za)*=β*+α*(1/zs)*α+β(1/zs)*,
dH(zs, za)=2dH(z1, z2),

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