Abstract

By analogy with the representation of the polarization of light on the Poincaré sphere, we describe the propagation and the reflection/transmission of light in a multilayer on a hyperbolic surface. We show that the propagation of light corresponds to a classical rotation on this surface and that its reflection/transmission corresponds to a hyperbolic rotation.

© 2002 Optical Society of America

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References

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  1. R. C. Jones, “A new calculus for treatment of optical system: I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  2. H. Hurwitz, R. C. Jones, “A new calculus for treatment of optical system: II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  3. R. C. Jones, “A new calculus for treatment of optical system: III. The Sohncke theory of optical activity,” J. Opt. Soc. Am. 31, 500–503 (1941).
    [CrossRef]
  4. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoı̈dales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).
  5. J.-M. Vigoureux, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
    [CrossRef]
  6. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
    [CrossRef]
  7. M. V. Berry, S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
    [CrossRef]
  8. J. J. Monzon, L. L. Sanchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [CrossRef]
  9. J. J. Monzon, L. L. Sanchez-Soto, “Fresnel formulas as Lorentz transformations,” J. Opt. Soc. Am. A 17, 1475–1481 (2000).
    [CrossRef]
  10. J.-M. Vigoureux, R. Giust, “The use of hyperbolic plane in studies of multilayers,” Opt. Commun. 186, 231–236 (2000).
    [CrossRef]
  11. N. Ya. Vilenkin, Fonctions spéciales et théorie de la représentation des groups (Dunod, Paris, 1969), Chap. VI.

2000

J.-M. Vigoureux, R. Giust, “The use of hyperbolic plane in studies of multilayers,” Opt. Commun. 186, 231–236 (2000).
[CrossRef]

J. J. Monzon, L. L. Sanchez-Soto, “Fresnel formulas as Lorentz transformations,” J. Opt. Soc. Am. A 17, 1475–1481 (2000).
[CrossRef]

1999

1998

J.-M. Vigoureux, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

1996

M. V. Berry, S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

1987

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1950

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoı̈dales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1941

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoı̈dales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Berry, M. V.

M. V. Berry, S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

Giust, R.

J.-M. Vigoureux, R. Giust, “The use of hyperbolic plane in studies of multilayers,” Opt. Commun. 186, 231–236 (2000).
[CrossRef]

Hurwitz, H.

Jones, R. C.

Klein, S.

M. V. Berry, S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

Monzon, J. J.

Sanchez-Soto, L. L.

Vigoureux, J.-M.

J.-M. Vigoureux, R. Giust, “The use of hyperbolic plane in studies of multilayers,” Opt. Commun. 186, 231–236 (2000).
[CrossRef]

J.-M. Vigoureux, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

Vilenkin, N. Ya.

N. Ya. Vilenkin, Fonctions spéciales et théorie de la représentation des groups (Dunod, Paris, 1969), Chap. VI.

Ann. Phys. (Paris)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoı̈dales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

J. Mod. Opt.

J.-M. Vigoureux, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

M. V. Berry, S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J.-M. Vigoureux, R. Giust, “The use of hyperbolic plane in studies of multilayers,” Opt. Commun. 186, 231–236 (2000).
[CrossRef]

Other

N. Ya. Vilenkin, Fonctions spéciales et théorie de la représentation des groups (Dunod, Paris, 1969), Chap. VI.

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Figures (5)

Fig. 1
Fig. 1

Decomposition of the electric field in forward and backward components at an interface.

Fig. 2
Fig. 2

Notation for the calculation of the Poynting vector of two interfering fields defined by the wave vectors kA and kB and the electric fields EA and EB.

Fig. 3
Fig. 3

Hyperboloid surface of two sheets described by the equation z2-x2-y2=1. The geometrical representation of the fields ei± needs only the positive sheet (z>0) when all the sources are positioned at the left of the multilayer.

Fig. 4
Fig. 4

Example of a path on the hyperboloid surface. The multilayer consists of three media (n1=1, n2=1.5, n3=2 and d1=100 nm, d2=301 nm). The wavelength is λ=0.9 μm.

Fig. 5
Fig. 5

Projection on the Oxy plane of the path of the state of light on the hyperboloid of Fig. 3 in the two cases where the electromagnetic field is the same before and behind a plane-parallel plate. (a) Three media with {n1, n2, n3}={1, 2, 1} refractive indices. The optical path of the light is equal to λ/2. (b) Three media with {n1, n2, n3}={1, 2, 4} refractive indices (n1n3=n22). The optical path of the light is equal to λ/2.

Equations (63)

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|Ψ=121i1-iExEy=Ex+iEy2Ex-iEy2.
x=Ψ|σx|Ψ=Ψ|0110|Ψ,
y=Ψ|σy|Ψ=Ψ|0-ii0|Ψ,
z=Ψ|σz|Ψ=Ψ|100-1|Ψ.
x2+y2+z2=1.
|φ(z)=E+(z)E-(z).
|φ(zj)=[βji]|φ(zi)=exp(iβji)00exp(-iβji)|φ(zi),
βji=kiz(zj-zi)=kizdji,
|φj=12kjzkjz+kizkjz-kizkjz-kizkjz+kiz|φi=1tji1rjirji1|φi=[Rji]|φi,
P=12μ0ω [|EA|2kA+|EB|2kB]+14μ0ω [EA  EB*+EA*  EB](kA+kB),
P1z=k1z2μ0ω [|E1+|2-|E1-|2].
P2z=k2z2μ0ω [|E2+|2-|E2-|2].
E2+=exp(iβ)E1+,
E2-=exp(-iβ)E1-.
E2+=1t21 E1++r21t21 E1-,
E2-=r21t21 E1++1t21 E1-,
ei±=kizEi±
I-= |ei+|2-|ei-|2,
|Φi=ei+ei-,
I-=Φi|σz|Φi.
[rji]=12kizkjzkjz+kizkjz-kizkjz-kizkjz+kiz .
xi=Φi|σx|Φi=[ei+*ei-+ei+ei-*],
yi=-Φi|σy|Φi=i[ei+*ei--ei+ei-*],
zi=Φi|σ0|Φi=|ei+|2+|ei-|2.
zi2-xi2-yi2=[|ei+|2-|ei-|2]2=I-2,
xi=Φi|σx|ΦiΦi|σz|Φi,
yi=-Φi|σy|ΦiΦi|σz|Φi,
zi=Φi|σ0|ΦiΦi|σz|Φi,
zi2=1+xi2+yi2.
|Φj=[βji]|Φi.
xj=Φj|σx|Φj/I-=Φi|[βji]σx[βji]|Φi/I-,
yj=-Φj|σy|Φj/I-=-Φi|[βji]σy[βji]|Φi/I-,
zj=Φj|σ0|Φj/I-=Φi|[βji]σ0[βji]|Φi/I-,
xj=xicos 2β-yisin 2β,
yj=xisin 2β+yicos 2β,
zj=zi.
xj=(kj2+ki2)xi+(kj2-ki2)zi2kikj,
yj=yi,
zj=(kj2-ki2)xi+(kj2+ki2)zi2kikj,
cosh 2θji=kj2+ki22kikj,
sinh 2θji=kj2-ki22kikj,
xj=(cosh 2θ)xi+(sinh 2θ)zi,
yj=yi,
zj=(sinh 2θ)xi+(cosh 2θ)zi.
rji=kj-kikj+ki=tanh θji.
sinh B=2r1-r2.
xC=-(sinh θ32)zD,
yC=yD,
zC=(cosh θ32)zD.
xB=(cos 2β)xC+(sin 2β)yC,
yB=-(sin 2β)xC+(cos 2β)yC,
zB=zC.
sinh 2θ32=-sinh 2θ21.
(n1-n3)(n1n3+n22)=0.
n cos θd=mλ/4,
sinh 2θ21=sinh 2θ32.
(n1+n3)(n1n3-n22)=0.
|Φ2=[M]|Φ1.
Φ2|σz|Φ2=Φ1|σz|Φ1,
[M]σz[M]=σz.
[M]=ρ exp(iα)ρ2-1exp[i(γ-δ+α)]ρ2-1exp(iδ)ρ exp(iγ),
 
[M]=ρ exp(iα)ρ2-1exp(-iδ)ρ2-1exp(iδ)ρ exp(-iα)=abb*a*,

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