Abstract

The extended Jones matrix method is applied to one dielectrically anisotropic, homogeneous thin film at oblique incidence. Standard boundary conditions are imposed on resultant electric- and magnetic-field vectors at interfaces. Thus simple matricial relations are obtained for transmitted and reflected electric-field amplitudes at the two interfaces. In the limits of isotropy, they reduce to four well-known Abelès relations, and thus they may be considered as generalized Abelès relations for dielectrically anisotropic thin films. These matricial relations include multiple reflections while dealing with total fields. Thus they provide new insights into the 2 × 2 extended Jones matrix formalism.

© 1997 Optical Society of America

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References

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  16. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
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  17. Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
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    [CrossRef]

1994

1993

S.-C. A. Lien, “Application of computer simulation to improve the optical performance of liquid crystal displays,” Opt. Eng. 32, 1762–1768 (1993).
[CrossRef]

C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
[CrossRef]

1992

1991

1990

Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
[CrossRef] [PubMed]

A. R. MacGregor, “Method for computing homogeneous liquid-crystal conoscopic figures,” J. Opt. Soc. Am. A 7, 337–347 (1990).
[CrossRef]

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

1985

1984

1982

1979

1972

1970

1948

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieus stratifiés,” Ann. Phys. 3, 504–520 (1948).

Abelès, F.

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieus stratifiés,” Ann. Phys. 3, 504–520 (1948).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7.

Berreman, D. W.

Fritsch, M.

Gu, C.

Haas, G.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 59–62.

Henvis, B. W.

Hodgkinson, I. J.

Horowitz, F.

Liang, Q.-T.

Lien, S.-C. A.

S.-C. A. Lien, “Application of computer simulation to improve the optical performance of liquid crystal displays,” Opt. Eng. 32, 1762–1768 (1993).
[CrossRef]

Lin-Chung, P. J.

Lu, K.

MacGregor, A. R.

Macleod, H. A.

Mansuripur, M.

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

Mavrudis, T.

Mentel, J.

Mlynski, D. A.

Saleh, B. E. A.

Schmidt, E.

Sikkens, M.

Teitler, S.

Wharton, J. J.

Wöhler, H.

Yeh, P.

Zhu, X.

Ann. Phys.

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieus stratifiés,” Ann. Phys. 3, 504–520 (1948).

Appl. Opt.

J. Appl. Phys.

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

S.-C. A. Lien, “Application of computer simulation to improve the optical performance of liquid crystal displays,” Opt. Eng. 32, 1762–1768 (1993).
[CrossRef]

Opt. Lett.

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.7.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, pp. 59–62.

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Figures (2)

Fig. 1
Fig. 1

Reflection and refraction of light incident on the plane x 2 - x 3 from the ambient region, Region 1, on an anisotropic film, Region 2, of thickness d, which is coated on a substrate, Region 3. Axis x 3 is normal to the film surfaces. The propagation directions of the incident, reflected, and forward-moving α and β waves are defined7 by wave vectors k 1 +, k 1 -, k α +, and k β +, respectively. The propagation directions of backward-moving α and β waves inside the film are defined by k α - and k β -. The direction of the refracted waves in the substrate is defined by k 3 +. In the upper left corner, 1, 2, and 3, are unit vectors parallel to the x 1, x 2, and x 3 axes.

Fig. 2
Fig. 2

Variation of R p against φ at two incident angles, θ1 = 30° and θ1 = 45°, for (a) d = λ/8 and (b) d = λ/4.

Equations (57)

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ki±=ω/cξxˆ2±ζixˆ3  i=1, 3,
kv±=ω/cξxˆ2+ζv±xˆ3  v=α, β,
¯=112113212223313233.
ζ4+q3ζ3+q2ζ2+q1ζ+q0=0,
q3=ξ23+32/33,
q2=ξ2-11-221-ξ2/33+1331+2332/33,
q1=ξ1231+1321+ξ2-1123+32/33,
q0=1-ξ2/332211-ξ2-1221+132132-2231+231231+32ξ2-11/33.
nv±=ξ2+ζv±21/2  v=α, β
θv±=arccosζv±/nv±  v=α, β.
kˆi±=0, sin θi, ±cos θiT  i=1, 3,
kˆv±=0, sin θv±, cos θv±T  v=α, β.
Region 1:  E1±=As±ŝ+Ap±pˆ1±expjωt-k1±r,
Region 2: E2±=Bα±êα± exp-jkα±r+Bβ±êβ± exp-jkβ±rexpjωt,
Region 3: E3+=Cs+ŝ+Cp+pˆ3+expjωt-k3+r,
pˆi±=kˆi±×ŝ=0, ±cos θi, -sin θiT  i=1, 3,
êv±=ev1±, ev2±, ev3±T,
ĥv±=hv1±, hv2±, hv3±T  v=α, β.
Region 1: H1±=k1±×E1±/ω,
Region 2: H2±=v=α,β Bv±kv±xêv±×exp-jkv±rexpjωt/ω,
Region 3: H3+=k3+×E3+/ω.
kvσ×êvσ=ĥvσkvσsinπ/2-δvσ=ĥvσnvσω/ccos δvσ.
H1±=n1/cAs±pˆ1±-Ap±ŝexpjωt-k1±r,
H2±=v=α,βnv±/cBv±ĥv± cos δv± exp-jkv±r×expjωt,
H3+=n3/cCs+pˆ3+-Cp+ŝexpjωt-k3+r.
av±iσ=ζiev1σ±nvσhv2σ cos δvσ,
bv±iσ=niev2σ±nvσhv1σ cos θi cos δvσ,
Di+=aα+i+bβ-i+-aβ+i+bα-i+,
Di-=aα-i-bβ+i--aβ-i-bα+i-,
τ111+τ121+τ211+τ221+As+Ap+=Bα+Bβ++ρ111+ρ121+ρ211+ρ221+Bα-Bβ-,
τ111-τ121-τ211-τ221-As-Ap-=ρ221--ρ121--ρ211-ρ111-Bα+Bβ++Bα-Bβ-,
-ρ223-ρ123-ρ213--ρ113-Xα+00Xα+Bα+Bβ+=Xα-00Xβ-Bα-Bβ-,
τ112+τ122+τ212+τ222+Xα+00Xβ+Bα+Bβ+=Cs+X3+Cp+X3+,
τ111±=2ζ1bβ1±/D1±,
τ121±=2ζ1aβ±1±/D1±,
τ211±=-2ζ1bα1±/D1±,
τ221±=±2ζ1aα±1±/D1±.
ρ11i±=aα±i-bβi+-aβ±i+bαi-/Di±  i=1, 3,
ρ12i±=aβ±i-bβi+-aβ±i+bβi-/Di±,
ρ21i±=aα±i+bαi--aα±i-bαi+/Di±,
ρ22i±=aα±i+bβi--aβ±i-bαi+/Di±.
τ112+=eα1+-eα1-ρ223-+eβ1-ρ213-,
τ122+=eβ1++eα1-ρ123--eβ1-ρ113-,
τ212+=nα-hα1-ρ223- cos δα--nβ-hβ1-ρ213- cos δβ--nα+hα1+ cos δα+/n3,
τ222+=nβ-hβ1-ρ113- cos δβ--nα-hα1-ρ123- cos δα--nβ+hβ1+ cos δβ+/n3,
Xv±=exp-jωζv±d/c  v=α, β,
X3+=exp-jωζ3d/c.
As-Ap-=rssrpsrpsrppAs+Ap+,
Cs+X3+Cp+X3+=tsstsptpstppAs+Ap+.
R=0.5rss2+rsp2+rps2+rpp2.
rss=aα-1+bβ-1+-aβ-1+bα-1+/D1+,
rsp=aα+1+aβ-1+-aβ+1+aα-1+/D1+,
rps=bα-1+bβ+1+-bα+1+bβ-1+/D1+,
rpp=aβ+1+bα+1+-aα+1+bβ+1+/D1+,
ζ2-o+ξ2Q2ζ2+2Q1ζ+Q0=0,
tanδe±=Δ tan γe±/e+o tan2 γe±,
rpp=τ221+/τ221-ρ111--ρ113- exp-j4πχd/λ/1-ρ221+ρ113- exp-j4πχd/λ,

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