Abstract

It is shown that the matrix describing a general (transparent or absorbing) multilayer is an element of the group SL(2, C), which is locally isomorphic to the (3+1)-dimensional restricted Lorentz group SO(3, 1). In consequence, a natural identification of the multilayer reflection and transmission coefficients with the parameters of a Lorentz transformation is performed. The correspondence between electric field and space–time variables is discussed as well.

© 1999 Optical Society of America

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References

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  1. L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).
  2. O. S. Heavens, Thin Film Physics (Methuen, London, 1970).
  3. H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, UK, 1986).
  4. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  6. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 249–288.
  7. 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]
  8. S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
    [CrossRef]
  9. I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
    [CrossRef]
  10. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  11. 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]
  12. A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991).
    [CrossRef]
  13. J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
    [CrossRef]
  14. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [CrossRef]
  15. A. O. Barut, R. Ra̧czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warsaw, 1977).
  16. V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
    [CrossRef]
  17. J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
    [CrossRef]
  18. J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) (to be published).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).
  20. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).
  21. A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980).
  22. D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  23. M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

1999 (1)

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

1998 (1)

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

1996 (1)

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

1993 (2)

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

1992 (1)

1991 (3)

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

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

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

1947 (1)

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

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 249–288.

Azzam, R. M. A.

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

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-Polish Scientific, Warsaw, 1977).

A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980).

Bashara, N. M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Brekovskikh, L. M.

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

Coldren, L. A.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Corzine, S. W.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

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]

Heavens, O. S.

O. S. Heavens, Thin Film Physics (Methuen, London, 1970).

Jackson, D. A.

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

Lekner, J.

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

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

Macleod, H. A.

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

Monzón, J. J.

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, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) (to be published).

Ra¸czka, R.

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

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

Sánchez-Soto, L. L.

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, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) (to be published).

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

Ungar, A. A.

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

Vigoureux, J. M.

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

Wigner, E. P.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Yan, R. H.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Am. J. Phys. (4)

J. J. Monzón, L. L. Sánchez-Soto, “On the concept of absorption for a Fabry–Perot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

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

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

I. V. Lindell, A. H. Sihvola, “The quotient function and its applications,” Am. J. Phys. 66, 197–202 (1998).
[CrossRef]

Ann. Math. (1)

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

IEEE J. Quantum Electron. (1)

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh-substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

J. Opt. Soc. Am. A (2)

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]

Other (13)

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

J. J. Monzón, L. L. Sánchez-Soto, “Algebraic structure of Fresnel reflection and transmission coefficients at an interface,” Optik (Stuttgart) (to be published).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

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

A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 1980).

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

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

O. S. Heavens, Thin Film Physics (Methuen, London, 1970).

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

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 249–288.

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Figures (1)

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.

Equations (60)

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

N0 sin θ0==Nj sin θj==Nm+1 sin θm+1.
Ea(-)=RasEa(+)+TsaEs(-),
Es(+)=TasEa(+)+RsaEs(-),
Ea(+)Ea(-)=1Tas1-RsaRasXEs(+)Es(-),
X=TasTsa-RasRsa.
Mas=I01L1I12L2I(j-1)jLjLmIm(m+1).
Iij=1tij1rijrij1,
det Iij=Nj cos θjNi cos θi,
Lj=exp(iδj)00exp(-iδj),
det Lj=1.
det Mas=Nm+1 cos θm+1N0 cos θ0=TsaTas.
W=abcd,
Tas=|Tas|exp(iτ),Ras=|Ras|exp(iρ),
Tsa=|Tsa|exp(iτ),Rsa=|Rsa|exp(iρ)
Ras=|Ras|2,Tas=Nm+1 cos θm+1N0 cos θ0 |Tas|2,
Rsa=|Rsa|2,Tsa=N0 cos θ0Nm+1 cos θm+1 |Tsa|2.
Ea(±)=1(N0 cos θ0)1/2 ea(±),
Es(±)=1(Nm+1 cos θm+1)1/2 es(±),
ea(+)ea(-)=1Tas exp(iτ)1-Rsa exp(iρ)Ras exp(iρ)TasTsa exp[i(τ+τ)]-RasRsa exp[i(ρ+ρ)]es(+)es(-).
xμ=Λνμxν
(x0)2-(x1)2-(x2)2-(x3)2
g=10000-10000-10000-1=(g)μν=(g)μν,
ΛTgΛ=g
det Λ=±1.
X=xμσμ=x0+x3x1-ix2x1+ix2x0-x3.
X=WXW,
Λνμ=12 tr(σμWσνW).
Λ=LR.
W=HU,
H=1(2+|a|2+|b|2+|c|2+|d|2)1/2×1+|a|2+|b|2ac*+bd*a*c+b*d1+|c|2+|d|21Γh1zz*h2,
U=1(2+|a|2+|b|2+|c|2+|d|2)1/2×a+d*b-c*-b*+ca*+du1u2-u2*u1*,
X=UXU=UXU-1,
R=10000Re(u12-u22)Im(u12+u22)-2 Re(u1u2)0-Im(u12-u22)Re(u12+u22)2 Im(u1u2)02 Re(u1u2*)2 Im(u1u2*)|u1|2-|u2|2.
R(αˆ, βˆ, γˆ)=cos αˆ cos βˆ cos γˆ-sin αˆ sin γˆ-cos αˆ cos βˆ sin γˆ-sin αˆ cos γˆcos αˆ sin βˆsin αˆ cos βˆ cos γˆ+cos αˆ sin γˆ-sin αˆ cosβˆ sin γˆ+cos αˆ cos γˆsin αˆ sin βˆ-sin βˆ cos γˆsin βˆ sin γˆcos βˆ.
αˆ=-arg(u1u2),
βˆ=arccos(|u1|2-|u2|2),
γˆ=-arg(u1u2*).
X=HXH=HXH,
L=Γ2/2-1Γ Re(z)-Γ Im(z)Γ/2Γ Re(z)1+2[Re(z)]2-2 Re(z)Im(z)Γ Re(z)/Γ-Γ Im(z)-2 Re(z)Im(z)1+2[Im(z)]2-Γ Im(z)/ΓΓ/2Γ Re(z)/Γ-Γ Im(z)/Γ1+Γ2/2Γ2,
Γ=|a|2+|b|2-|c|2-|d|2.
L=γ-γβ1-γβ2-γβ3-γβ11+Aβ12Aβ1β2Aβ1β3-γβ2Aβ2β11+Aβ22Aβ2β3-γβ3Aβ3β1Aβ3β21+Aβ32,
γ=11-β2,A=γ-1β2.
γ=Γ2/2-1=(|a|2+|b|2+|c|2+|d|2)/2,
γβ1=-Γ Re(z),
γβ2=Γ Im(z),
γβ3=-Γ/2.
e=e(+)e(-),
E=ee=|e(+)|2e(+)e(-)*e(+)*e(-)|e(-)|2,
Ea=MasEsMas,
x0x1x2x3[|es(+)|2+|es(-)|2]/2Re[es(+)*es(-)]Im[es(+)*es(-)][|es(+)|2-|es(-)|2]/2,
(x0)2-(x1)2-(x2)2-(x3)2
=(x0)2-(x1)2-(x2)2-(x3)2=0,
Tas=T=|T|exp(iτ),Ras=R=|R|exp(iρ),
T=|T|2,R=|R|2.
Wlossless=1/TR*/T*R/T1/T*=1|T|1|R|exp(-iρ)|R|exp(iρ)1
×exp(-iτ)00exp(iτ)=HU,
T+R+A=1.
Wsymm=1T1-RRT2-R2=1|T|1|R|exp(-iρ)|R|exp(iρ)1-A
×exp(-iτ)00exp(iτ)=HU.
β3=-Γ2γ=A(A-2)2γ T,

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