Abstract

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in terms of three basic matrices of simple interpretation. By considering the bilinear transformation that this transfer matrix induces in the complex plane, we introduce the concept of multilayer transfer function and study its properties in the unit disk. In this geometrical setting, our factorization translates into three actions that can be viewed as the basic components for understanding the multilayer behavior. Additionally, we introduce a simple trace criterion that allows us to classify multilayers into three types with properties closely related to one (and only one) of these three basic matrices. We apply this approach to analyze some practical examples that are typical of these types of matrices.

© 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 (Kluwer Academic, Dordrecht, The Netherlands, 1987).
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    [CrossRef]
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    [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, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [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. Sel. Top. Quantum Electron. 34, 1075–1081 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. When ambient (0) and substrate (m+1) media are different, the angles θ0 and θm+1 are connected by Snell’s law, n0 sin θ0=nm+1 sin θm+1, where nj denotes the refractive index of the jth medium.
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    [CrossRef]
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    [CrossRef]
  24. Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
    [CrossRef]
  25. J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1989).
    [CrossRef]
  26. J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
    [CrossRef]
  27. A. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986).
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    [CrossRef]
  29. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999) Sect. 1.6.5.
  30. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A 11, 2892–2899 (1994).
    [CrossRef]
  31. J. Lekner, “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A Pure Appl. Opt. 2, 349–352 (2000).
    [CrossRef]
  32. H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
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    [CrossRef]
  35. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  36. 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]

2002 (1)

2001 (3)

2000 (1)

J. Lekner, “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A Pure Appl. Opt. 2, 349–352 (2000).
[CrossRef]

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. Sel. Top. Quantum Electron. 34, 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]

1995 (1)

1994 (2)

1990 (1)

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

1989 (2)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1989).
[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 (2)

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

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[CrossRef]

1980 (1)

V. Degiorgio, “Phase shift between the transmitted and reflected optical fields of a semireflecting lossless mirror is π/2,” Am. J. Phys. 48, 81–82 (1980).
[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.

Arsenault, H. H.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

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]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999) Sect. 1.6.5.

Cadilhac, M.

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

Cariñena, J. F.

Cohen, N.

Coxeter, H. S. M.

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

Degiorgio, V.

V. Degiorgio, “Phase shift between the transmitted and reflected optical fields of a semireflecting lossless mirror is π/2,” Am. J. Phys. 48, 81–82 (1980).
[CrossRef]

Franta, D.

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

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. Sel. Top. Quantum Electron. 34, 1075–1081 (1998).
[CrossRef]

Lekner, J.

J. Lekner, “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A Pure Appl. Opt. 2, 349–352 (2000).
[CrossRef]

J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A 11, 2892–2899 (1994).
[CrossRef]

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1989).
[CrossRef]

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

López-Lacasta, C.

Macleod, H. A.

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

Macukow, B.

Mandel, L.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Monzón, J. J.

Mukunda, N.

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. Sel. Top. Quantum Electron. 34, 1075–1081 (1998).
[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), p. 181.

Ou, Z. Y.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Perelomov, A.

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

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. Sel. Top. Quantum Electron. 34, 1075–1081 (1998).
[CrossRef]

Sánchez-Soto, L. L.

Shamir, J.

Sheridan, J. T.

Simon, R.

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. Sel. Top. Quantum Electron. 34, 1075–1081 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999) Sect. 1.6.5.

Yeh, P.

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

Yonte, T.

Zeilinger, A.

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[CrossRef]

Am. J. Phys. (4)

V. Degiorgio, “Phase shift between the transmitted and reflected optical fields of a semireflecting lossless mirror is π/2,” Am. J. Phys. 48, 81–82 (1980).
[CrossRef]

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[CrossRef]

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

J. Lekner, “The phase relation between reflected and transmitted waves, and some consequences,” Am. J. Phys. 58, 317–320 (1990).
[CrossRef]

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]

Can. J. Phys. (1)

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1989).
[CrossRef]

IEEE J. Sel. Top. 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. Sel. Top. Quantum Electron. 34, 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. A Pure Appl. Opt. (1)

J. Lekner, “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A Pure Appl. Opt. 2, 349–352 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

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. (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[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. (3)

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]

Phys. Rev. A (2)

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]

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

Other (10)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999) Sect. 1.6.5.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

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 (Kluwer Academic, Dordrecht, The Netherlands, 1987).

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

When ambient (0) and substrate (m+1) media are different, the angles θ0 and θm+1 are connected by Snell’s law, n0 sin θ0=nm+1 sin θm+1, where nj denotes the refractive index of the jth medium.

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

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

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

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Figures (5)

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

Plot of several orbits in the unit disk of the elements of the Iwasawa decomposition K(ϕ), A(ξ), and N(ν) for the group of multilayer transfer matrices.

Fig. 3
Fig. 3

Geometrical representation in the unit disk of the action of 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 phase thickness β2=3 rad. The point zs=0.4 exp(-iπ/3) is transformed by the system into the point za=-0.44+0.49i. We indicate the three orbits given by the Iwasawa decomposition and, as a thick line, the trajectory associated with the multilayer action.

Fig. 4
Fig. 4

Plot of the values of [Tr(Mas)]2 and of the parameters ϕ, ξ, and ν in the Iwasawa decomposition for the same system as in Fig. 3, as a function of β2.

Fig. 5
Fig. 5

Same plot as in Fig. 4 but for an asymmetric system made up of a glass plate (n1=1.5 and β1=2.75 rad) coated with a zinc sulfide film (n2=2.3).

Equations (40)

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),
TasTsa-RasRsa=exp(i2τ),Rsa=-Ras* exp(i2τ),
Mas-1=Msa*,
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,
ρ-τ=±π/2.
ν=[exp(ξ)-1]tan(ϕ/2);
Ras=exp(-iϕ)tanh(ξ/2)[tan(ϕ/2)-i]1-i tanh(ξ/2) tan(ϕ/2).
ξ=0,
dRasdl=Rasϕξ=0dϕdl+Rasξξ=0dξdl=0.
dξdl=0.
z=E(-)/E(+),
za=Φ[Mas, zs]=β*+α*zsα+βzs.
z=Φ[Mas, z],
z=-i Im(α)±{[Re(α)]2-1}1/2β.
Tr(Mas)=2 Re(α)=2 cos τ|Tas|,
Mˆas=CMasC-1,
C=c1c2c2*c1*,
αˆ=α|c1|2-α*|c2|2-2i Im(βc1c2*),
βˆ=βc12-β*c22-2ic1c2 Im(α).
Φ[CMasC-1, 0]=Φ[CMas, zf]=Φ[C, zf]=0.
c1=1(1-|zf|2)1/2exp(iδ),c2=-c1zf*,
Mˆas(μ1)Mˆas(μ2)=Mˆas(μ1+μ2),
MasN=C-1MˆasN(μ)C=C-1Mˆas(Nμ)C,
U=121ii1.
Ea=MasEs,
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.
TasT010=(1-r012)exp(-iβ1)1-r012 exp(-i2β1),
[Tr(M010)]2=4 cos2 β14,

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