Abstract

A wide-angle beam propagation method suitable for analyzing anisotropic devices involving liquid crystals is presented. The mathematical formulation is based on a system of coupled differential equations involving an electric and a magnetic field component. The contribution of all dielectric tensor elements is included. A numerical implementation based on finite differences is used. Numerical examples are focused on light-wave propagation within twisted nematic pixels found in microdisplays, with all effects arising at pixel edges that are incorporated. A comparison between the results obtained and the prediction of finite-difference time-domain simulations is conducted, showing satisfactory agreement. The required computational effort is found to be minimal.

© 2000 Optical Society of America

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References

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  1. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  2. B. Witzigmann, P. Regli, W. Fichtner, “Rigorous electromagnetic simulation of liquid crystal displays,” J. Opt. Soc. Am. A 15, 753–757 (1998).
    [CrossRef]
  3. E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).
  4. E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
    [CrossRef]
  5. L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [CrossRef] [PubMed]
  6. J. M. Liu, L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 1574–1585 (1992).
    [CrossRef]
  7. C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
    [CrossRef]
  8. F. Castaldo, G. Abbate, E. Santamato, “Theory for a new full-vectorial beam-propagation method in anisotropic structures,” Appl. Opt. 38, 3904–3910 (1999).
    [CrossRef]
  9. Y. Tsuji, M. Koshiba, N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. 17, 723–728 (1999).
    [CrossRef]
  10. E. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).
  11. Y. Ohkawa, Y. Tsuji, M. Koshiba, “Analysis of anisotropic dielectric grating diffraction using the finite-element method,” J. Opt. Soc. Am. A 13, 1006–1012 (1996).
    [CrossRef]
  12. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef]
  13. W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
    [CrossRef]
  14. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
    [CrossRef]
  15. W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

2000

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

1999

1998

1996

1994

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

1992

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

J. M. Liu, L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 1574–1585 (1992).
[CrossRef]

G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
[CrossRef]

1982

1972

Abbate, G.

Berreman, D. W.

Castaldo, F.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Elston, S. J.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).

E. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).

Fichtner, W.

Filippov, S. K.

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).

Gomelsky, L.

Hadley, G. R.

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
[CrossRef]

Huang, W. P.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Koshiba, M.

Kriezis, E. E.

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).

Liu, J. M.

Ohkawa, Y.

Press, W. H.

W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Regli, P.

Santamato, E.

Takimoto, N.

Thylen, L.

Tsuji, Y.

Witzigmann, B.

Xu, C. L.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Yevick, D.

Appl. Opt.

IEEE J. Quantum Electron.

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 363–370 (1992).
[CrossRef]

J. Lightwave Technol.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Y. Tsuji, M. Koshiba, N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. 17, 723–728 (1999).
[CrossRef]

J. Mod. Opt.

E. E. Kriezis, S. J. Elston, “A wide angle beam propagation method for the analysis of tilted nematic liquid crystal structures,” J. Mod. Opt. 46, 1201–1212 (1999).

J. Opt.

E. E. Kriezis, S. K. Filippov, S. J. Elston, “Light propagation in domain walls in ferroelectric liquid crystal devices by the finite-difference time-domain method,” J. Opt. 2, 27–33 (2000).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal displays by the finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Opt. Lett.

Other

W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of a two-dimensional device showing the various conventions used for the measurement of the angles involved. PBC, periodic boundary condition. k inc, incident wave vector.

Fig. 2
Fig. 2

Application example geometry representing a 25-µm-wide by 5-µm-thick TN microdisplay pixel. The driving electrodes are shown together with an intergap of 10 µm. The figure corresponds to a single period of the model used and is composed of half of the electrode width associated with the pixel to the left of the intergap, the intergap, and half of the electrode associated with the pixel to the right. ITO, indium tin oxide.

Fig. 3
Fig. 3

Normalized transmitted near-field optical intensity plotted versus the distance along the cell ( axis) for different applied static voltages (2, 3, 4, and 5 V). The curves shown were calculated by the proposed BPM and the FDTD method. Illumination is at normal incidence.

Fig. 4
Fig. 4

Normalized transmitted near-field optical intensity plotted versus the distance along the cell ( axis) for different angles of incidence. A static voltage of 4 V was applied to the electrodes. The curves shown were calculated by the proposed BPM and the FDTD method. (a) θinc = ±15°. (b) θinc = ±30°. Angles of incidence are measured within the lower supporting glass plate.

Fig. 5
Fig. 5

Plane-wave spectrum representation of the field components along the ŷ direction. The amplitude of each plane-wave component is plotted versus its angle of propagation. (a) E y (BPM) at θinc = 0°, (b) H y (BPM) at θinc = 0°, (c) E y (FDTD) at θinc = 0°, (d) H y (FDTD) at θinc = 0°, (e) E y (BPM) at θinc = 30°, (f) H y (BPM) at θinc = 30°, (g) E y (FDTD) at θinc = 30°, (h) H y (FDTD) at θinc = 30°.

Tables (1)

Tables Icon

Table 1 Comparison between the Execution Time and the Memory Requirements for the BPM and the FDTD Methodsa

Equations (35)

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

-Eyz=-jωμ0Hx,
-Ezx+Exz=-jωμ0Hy,
Eyx=-jωμ0Hz,
-Hyz=jωε0εxxEx+εxyEy+εxzEz,
-Hzx+Hxz=jωε0εyxEx+εyyEy+εyzEz,
Hyx=jωε0εzxEx+εzyEy+εzzEz.
Ex=-1jωε0aεzzHyz+εxzHyx-ca Ey,
Ez=1jωε0aεxxHyx+εzxHyz-ba Ey,
a=εxxεzz-εzxεxz, b=εxxεzy-εzxεxy, c=εxyεzz-εzyεxz.
Hx=1jωμ0Eyz,
Hz=-1jωμ0Eyx.
2Eyx2+2Eyz2+k02εyy-εyzba-εyxcaEy-jωμ0ab˜ Hyx-c˜ Hyz=0,
jωε0zca-xbaEy+jωε0caEyz-baEyx+εxxa2Hyx2+εzza2Hyz2+2 εxza2Hyxz+xεxxa+zεxzaHyx+xεzxa+zεzzaHyz+k02Hy=0,
b˜=εyzεxx-εyxεxz, c˜=εyxεzz-εyzεzx.
Eyx, z=yx, zexp-jkrefz,
Hyx, z=yx, zexp-jkrefz.
100q2z2yy+R11R12R21R22+000S22/x×zyy+A11A12A21A22yy=0.
z-A100qz+R+S.
zn+1=-A100qzn+R+S, z0=0.
R˜+Szyy=-Ayy,
R˜=R-100qR-1A.
R˜r+1/2+Sr+1/2+αδzAr+1/2Ψr+1=R˜r+1/2+Sr+1/2+α-1δzAr+1/2Ψr.
Rr+1/2+Sr+1/2+Cr+1/2αAr+1/2Ψr+1=Rr+1/2+Sr+1/2+Cr+1/2α-1Ar+1/2Ψr,
Cα=αδz-1|R| R221|R| R12q|R| R21αδz-1|R| R11.
H11H12H21H22yr+1yr+1=B1B2.
yL, z=y0, zexp-jk0nglass sinθinc,
yL, z=y0, zexp-jk0nglass sinθinc.
ε˜x, z=no2+Δεr cos2 θ cos2 ϕΔεr cos2 θ sin ϕ cos ϕΔεr sin θ cos θ cos ϕΔεr cos2 θ sin ϕ cos ϕno2+Δεr cos2 θ sin2 ϕΔεr sin θ cos θ sin ϕΔεr sin θ cos θ cos ϕΔεr sin θ cos θ sin ϕno2+Δεr sin2 θ, Δεr=ne2-no2.
R11R12R21R22=-2jkrefjωμ0c˜ajωε0ca-2jkrefεzza+xεzxa+zεzza.
A11A12A21A22=P11P12P21P222x2+Q11Q12Q21Q22x+T11T12T21T22,
P=100εxxa,
Q=0-jωμ0b˜a-jωε0ba-2jkrefεxza+xεxxa+zεxza,
T=k02εyy-εyzba-εyxca-kref2krefωε0ca+jωε0-xba+zcakrefωμ0c˜a k02-εzza kref2-jkrefxεzxa+zεzza.
q=εzza,
S22=2 εxza.

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