Abstract

A new optical element that displays singular polarization eigenstates is proposed. It consists of a planar stratified structure composed of alternate gyrotropic and birefringent layers. The orthogonality of the polarization eigenstates is lost because of anisotropic reflections at the interfaces, which are enhanced by the special condition chosen for the multiple-beam interference. First we show that the anisotropic reflection at the interface between the layers with linear and circular symmetries does produce strong enough dichroism to break the orthogonality of polarization eigenstates. Second, we investigate the behavior of these eigenstates with respect to their linearity and orthogonality as a function of the width of the layers. Our results concretely demonstrate that it is possible to control the effective optical parameters of such stratified structures by adjusting the thickness of each anisotropic layer. Finally, we obtain the necessary conditions for designing a double-layer system with singular eigenstates of linear polarization.

© 2000 Optical Society of America

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References

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  1. S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. 48, 280–302 (1958).
  2. A. I. Okorochkov, A. F. Konstantantinova, “Influence of non-orthogonal characteristic waves in a crystal on the polarization of transmitted light,” Sov. Phys. Crystallogr. 30, 57–62 (1985).
  3. B. N. Grechushnikov, A. F. Konstantantinova, “Crystal optics of absorbing and gyrotropic media,” Comput. Math. Appl. 16, 637–655 (1988).
    [CrossRef]
  4. A. F. Konstantinova, E. A. Evdischenko, “Some methods for determination of optical parameters in gyrotropic crystals,” in Polarimetry and Ellipsometry, K. Dolny, M. Pluta, R. Wolinski, eds., Proc. SPIE3094, 159–168 (1997).
    [CrossRef]
  5. I. Prikryl, “Effect of disk birefringence on a differential magneto-optic readout,” Appl. Opt. 31, 1853–1862 (1992).
    [CrossRef] [PubMed]
  6. Y. C. Hsieh, M. Mansuripur, “Image contrast in polarization microscopy of magneto-optical disk data-storage media through birefringent plastic substrates,” Appl. Opt. 36, 4839–4852 (1997).
    [CrossRef] [PubMed]
  7. U. A. Leitão, L. C. Meira-Belo are preparing a manuscript to be called, “Non-orthogonal eigenstates of polarization in systems with superposition of optical effects.”
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).
  9. H. Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).
  10. K. Hoffman, R. Kunze, Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1971).
  11. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  12. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  13. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  14. There is also γ(ϕ0, ψ) = 0 in the regions where the oscillatory behavior occurs; see Fig. 2. The optical behavior in these points are rather trivial and will be not discussed.

1997 (1)

1992 (1)

1988 (1)

B. N. Grechushnikov, A. F. Konstantantinova, “Crystal optics of absorbing and gyrotropic media,” Comput. Math. Appl. 16, 637–655 (1988).
[CrossRef]

1985 (1)

A. I. Okorochkov, A. F. Konstantantinova, “Influence of non-orthogonal characteristic waves in a crystal on the polarization of transmitted light,” Sov. Phys. Crystallogr. 30, 57–62 (1985).

1972 (1)

1970 (1)

1958 (1)

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. 48, 280–302 (1958).

1948 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Berreman, D. W.

Evdischenko, E. A.

A. F. Konstantinova, E. A. Evdischenko, “Some methods for determination of optical parameters in gyrotropic crystals,” in Polarimetry and Ellipsometry, K. Dolny, M. Pluta, R. Wolinski, eds., Proc. SPIE3094, 159–168 (1997).
[CrossRef]

Grechushnikov, B. N.

B. N. Grechushnikov, A. F. Konstantantinova, “Crystal optics of absorbing and gyrotropic media,” Comput. Math. Appl. 16, 637–655 (1988).
[CrossRef]

Henvis, B. W.

Hoffman, K.

K. Hoffman, R. Kunze, Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Hsieh, Y. C.

Jones, R. C.

Konstantantinova, A. F.

B. N. Grechushnikov, A. F. Konstantantinova, “Crystal optics of absorbing and gyrotropic media,” Comput. Math. Appl. 16, 637–655 (1988).
[CrossRef]

A. I. Okorochkov, A. F. Konstantantinova, “Influence of non-orthogonal characteristic waves in a crystal on the polarization of transmitted light,” Sov. Phys. Crystallogr. 30, 57–62 (1985).

Konstantinova, A. F.

A. F. Konstantinova, E. A. Evdischenko, “Some methods for determination of optical parameters in gyrotropic crystals,” in Polarimetry and Ellipsometry, K. Dolny, M. Pluta, R. Wolinski, eds., Proc. SPIE3094, 159–168 (1997).
[CrossRef]

Kunze, R.

K. Hoffman, R. Kunze, Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Lang, H.

H. Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

Leitão, U. A.

U. A. Leitão, L. C. Meira-Belo are preparing a manuscript to be called, “Non-orthogonal eigenstates of polarization in systems with superposition of optical effects.”

Mansuripur, M.

Meira-Belo, L. C.

U. A. Leitão, L. C. Meira-Belo are preparing a manuscript to be called, “Non-orthogonal eigenstates of polarization in systems with superposition of optical effects.”

Okorochkov, A. I.

A. I. Okorochkov, A. F. Konstantantinova, “Influence of non-orthogonal characteristic waves in a crystal on the polarization of transmitted light,” Sov. Phys. Crystallogr. 30, 57–62 (1985).

Pancharatnam, S.

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. 48, 280–302 (1958).

Prikryl, I.

Teitler, S.

Appl. Opt. (2)

Comput. Math. Appl. (1)

B. N. Grechushnikov, A. F. Konstantantinova, “Crystal optics of absorbing and gyrotropic media,” Comput. Math. Appl. 16, 637–655 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. 48, 280–302 (1958).

Sov. Phys. Crystallogr. (1)

A. I. Okorochkov, A. F. Konstantantinova, “Influence of non-orthogonal characteristic waves in a crystal on the polarization of transmitted light,” Sov. Phys. Crystallogr. 30, 57–62 (1985).

Other (6)

A. F. Konstantinova, E. A. Evdischenko, “Some methods for determination of optical parameters in gyrotropic crystals,” in Polarimetry and Ellipsometry, K. Dolny, M. Pluta, R. Wolinski, eds., Proc. SPIE3094, 159–168 (1997).
[CrossRef]

U. A. Leitão, L. C. Meira-Belo are preparing a manuscript to be called, “Non-orthogonal eigenstates of polarization in systems with superposition of optical effects.”

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

H. Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

K. Hoffman, R. Kunze, Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1971).

There is also γ(ϕ0, ψ) = 0 in the regions where the oscillatory behavior occurs; see Fig. 2. The optical behavior in these points are rather trivial and will be not discussed.

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Figures (7)

Fig. 1
Fig. 1

Light-wave vectors and fields for a monochromatic light beam with normal incidence upon a transparent dielectric stratified structure.

Fig. 2
Fig. 2

Phase difference γ(ϕ0, ψ) for one eigenvector as a function of ψ about a critical value ψ m , γ(ϕ0, ψ) versus (ψ - ψ m ). The function jumps discontinuously from negative to positive values in most cases. Nevertheless, at ψ m the passage through zero is continuous, as shown in the inset. See text.

Fig. 3
Fig. 3

Difference between the azimuthal angles of the two eigenvectors, θ+ - θ-, as a function of ψ about the critical value ψ m , (θ+ - θ-) versus (ψ - ψ m ). The inset shows in detail the region near ψ m .

Fig. 4
Fig. 4

Phase difference γ(ϕ, ψ m ) as a function of ϕ in integer steps of the isotropic term’s short period, γ(ϕ, ψ m ) versus ϕ. Inset, the second linear region (i = 1; see text). The double arrow shows the width of the linear region (∼Δϕ = 3020 rad, yielding Δd 1 ≈ 300 µm; see text).

Fig. 5
Fig. 5

Phase difference γ(ϕ, ψ m ) versus ϕ (solid curves) and the difference of the eigenvectors’ azimuthal angles θ+ - θ- versus ϕ (dashed curves). Open circles, the linear solutions γ(ϕ i,j , ψ m ) = 0; filled circles, the corresponding values of θ+ - θ-; see text. Dotted lines show the discontinuity of γ(ϕ i,j , ψ m ).

Fig. 6
Fig. 6

(a) Azimuthal angle θ(ϕ i,j , ψ m ) and (b) ellipticity angle ∊(ϕ i,j , ψ m ) as functions of Δϕ = (ϕ i,j - ϕ i,0) about a linear region. Solid and dashed curves are parts of each of both eigenvectors; see text. Note that the ellipticities of both eigenvectors are identically null within the central region. Otherwise, the azimuths assume coincident values out of this region.

Fig. 7
Fig. 7

Intensity of transmitted light of a double-plate system between crossed polarizers for verification of the presence of degenerate behavior: (a) orthogonal PE’s (j = 0, center of the linear region), (b) nonorthogonal PE’s (j = 370, approximately halfway to the linear region border), (c) SPE’s (j = j max, on the linear region border). The corresponding PE’s are displayed above.

Tables (1)

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Table 1 Optical parameters and Orientations of the Transparent Plates Used in the Simulations

Equations (59)

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E0=a0xa0y,
E0=a0xa0y,
E1=al21i+ar21-i,
E1=al21i+ar21-i,
E2=a2xa2y,
E2=a2xa2y,
ET=aTxaTy,
a0xa0y+a0xa0y=al+al21i+ar+ar21-i.
n0a0xa0y-n0a0xa0y=nlal-al21i+nrar-ar21-i.
ϕ=2πλ0 d1,
12al expinlϕ+al exp-inlϕ1i+12×ar expinrϕ+ar exp-inrϕ1-i=a2xa2y+a2xa2y.
nl2al expinlϕ-al exp-inlϕ1i+nr2×ar expinrϕ-ar exp-inrϕ1-i=nxa2x-a2xnxa2y-a2y.
ψ=2πλ0 d2.
a2x expinxψ+a2x exp-inxψa2y expinyψ+a2y exp-inyψ=aTxaTy,
nxa2x expinxψ-a2x exp-inxψnya2y expinyψ+a2y exp-inyψ=nTaTxaTy.
AE0+AE0=ET,
BE0+BE0=ET.
E0=B-A-1A-BE0=RE0,
R=B-A-1A-B.
AE0+AE0=ET=TE0,
A+ARE0=TE0,
T=A+AR.
T·E0T=λE0T,
R·E0R=λE0R,
E0T=a0xexpiϕxa0yexpiϕy
γ=½ϕx-ϕy.
=arctanb/a,
Δψ=2πnx-ny.
ψm=2πnx-nyQ+Pm,
Δϕ1=2πnr+nl=2.034 rad,
Δϕ2=2πnr-nl=8.976×104 rad,
ϕi,j=ϕ0+2πnr-nl i+2πnr+nl j,
Q=ig02+ω2-p02+δ2+2ip0g0-ωδ1/2,
g02+ω2-p02-δ2=0,
p0g0-ωδ=0.
A=a1c1a3c2a4c1a2c2,
A=a1c1a3c2a4c1a2c2,
B=b1d1b3d2b4d1b2d2,
B=b1d1b3d2b4d1b2d2,
a1=cosnlϕ+i n0nl sinnlϕ+cosnrϕ+i n0nr sinnrϕ,
a1=cosnlϕ-i n0nl sinnlϕ+cosnrϕ-i n0nr sinnrϕ,
a2=cosnlϕ+i n0nl sinnlϕ+cosnrϕ+i n0nr sinnrϕ,
a2=cosnlϕ-i n0nl sinnlϕ+cosnrϕ-i n0nr sinnrϕ,
a3=i cosnlϕ-n0nl sinnlϕ-i cosnrϕ+n0nr sinnrϕ,
a3=i cosnlϕ+n0nl sinnlϕ-i cosnrϕ-n0nr sinnrϕ,
a4=-i cosnlϕ+n0nl sinnlϕ+i cosnrϕ-n0nr sinnrϕ,
a4=-i cosnlϕ-n0nl sinnlϕ+i cosnrϕ+n0nr sinnrϕ,
b1=n0 cosnlϕ+inl sinnlϕ+n0 cosnrϕ+inr sinnrϕ,
b1=-n0 cosnlϕ+inl sinnlϕ-n0 cosnrϕ+inr sinnrϕ,
b2=n0 cosnlϕ+inl sinnlϕ+n0 cosnrϕ+inr sinnrϕ,
b2=-n0 cosnlϕ+inl sinnlϕ-n0 cosnrϕ+inr sinnrϕ,
b3=in0 cosnlϕ-nl sinnlϕ-in0 cosnrϕ+nr sinnrϕ,
b3=-in0 cosnlϕ-nl sinnlϕ+in0 cosnrϕ+nr sinnrϕ,
b4=-in0 cosnlϕ+nl sinnlϕ+in0 cosnrϕ+nr sinnrϕ,
b4=in0 cosnlϕ+nl sinnlϕ-in0 cosnrϕ+nr sinnrϕ,
c1=2cosnxψ-i ntnx sinnxψ,
c1=2cosnyψ-i ntny sinnyψ,
d1=2nxnTnx cosnxψ-i sinnxψ,
d1=2nynTny cosnyψ-i sinnyψ.

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