Abstract

Cholesteric liquid crystals possessing a pitch gradient are expected to occur in several circumstances of some interest. It is necessary to investigate a layer’s light-reflection properties in order to confirm this structure and to further study its response to boundary or field conditions. In this article two different models and their solutions are presented.

© 1979 Optical Society of America

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References

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  1. A. C. Eringen and J. D. Lee, Liquid Crystals and Ordered Fluids, Vol. 2, edited by J. F. Johnson and R. S. Porter (Plenum, New York, 1974), pp. 383–401.
    [CrossRef]
  2. A. C. Neville and S. Caveney, “Scarabaeid Beetle Exocuticle as an Optical Analogue of Cholesteric Liquid Crystals,” Biol. Rev. 44, 531–562, (1969).
    [CrossRef] [PubMed]
  3. A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
    [CrossRef]
  4. L. E. Hajdo and A. C. Eringen, “A Theory of Light Reflection by Cholesteric Liquid Crystals Possessing a Tilted Structure,” (unpublished).
  5. G. H. Conners, “Electromagnetic wave propagation in cholesteric materials,” J. Opt. Soc. Am. 58, 875–879, (1968).
    [CrossRef]
  6. L. M. Brekhovskikh, “Waves in Layered Media” (Academic, New York, 1960).
  7. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 4, 502–510 (1972).
    [CrossRef]
  8. D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: Application to liquid-crystal twist cells,” J. Opt. Soc. Am. 61, 11, 1374–1380, (1973).
  9. R. Nityanda, “On the Theory of Light Propagation in Cholesteric Liquid Crystals,” Mol. Cryst. Liq. Cryst. 21, 315–331, (1973).
    [CrossRef]
  10. S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
    [CrossRef]
  11. R. C. Jones, wrote a series of eight noteworthy papers on optics, the last of which is in J. Opt. Soc. Am. 46, 126, (1956).
    [CrossRef]
  12. S. Chandrasekhar and K. N. S. Rao “Optical Rotatory Power of Liquid Crystals,” Acta. Crystallogr. A 24, 445–451, (1968).
    [CrossRef]

1976 (1)

S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
[CrossRef]

1974 (1)

A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
[CrossRef]

1973 (2)

D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: Application to liquid-crystal twist cells,” J. Opt. Soc. Am. 61, 11, 1374–1380, (1973).

R. Nityanda, “On the Theory of Light Propagation in Cholesteric Liquid Crystals,” Mol. Cryst. Liq. Cryst. 21, 315–331, (1973).
[CrossRef]

1972 (1)

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 4, 502–510 (1972).
[CrossRef]

1969 (1)

A. C. Neville and S. Caveney, “Scarabaeid Beetle Exocuticle as an Optical Analogue of Cholesteric Liquid Crystals,” Biol. Rev. 44, 531–562, (1969).
[CrossRef] [PubMed]

1968 (2)

G. H. Conners, “Electromagnetic wave propagation in cholesteric materials,” J. Opt. Soc. Am. 58, 875–879, (1968).
[CrossRef]

S. Chandrasekhar and K. N. S. Rao “Optical Rotatory Power of Liquid Crystals,” Acta. Crystallogr. A 24, 445–451, (1968).
[CrossRef]

1956 (1)

Berreman, D. W.

D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: Application to liquid-crystal twist cells,” J. Opt. Soc. Am. 61, 11, 1374–1380, (1973).

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 4, 502–510 (1972).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, “Waves in Layered Media” (Academic, New York, 1960).

Caveney, S.

A. C. Neville and S. Caveney, “Scarabaeid Beetle Exocuticle as an Optical Analogue of Cholesteric Liquid Crystals,” Biol. Rev. 44, 531–562, (1969).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar and K. N. S. Rao “Optical Rotatory Power of Liquid Crystals,” Acta. Crystallogr. A 24, 445–451, (1968).
[CrossRef]

Conners, G. H.

Eringen, A. C.

L. E. Hajdo and A. C. Eringen, “A Theory of Light Reflection by Cholesteric Liquid Crystals Possessing a Tilted Structure,” (unpublished).

A. C. Eringen and J. D. Lee, Liquid Crystals and Ordered Fluids, Vol. 2, edited by J. F. Johnson and R. S. Porter (Plenum, New York, 1974), pp. 383–401.
[CrossRef]

Hajdo, L. E.

L. E. Hajdo and A. C. Eringen, “A Theory of Light Reflection by Cholesteric Liquid Crystals Possessing a Tilted Structure,” (unpublished).

Jones, R. C.

Kizior, G. J.

A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
[CrossRef]

Kozinski, A. A.

A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
[CrossRef]

Lee, J. D.

A. C. Eringen and J. D. Lee, Liquid Crystals and Ordered Fluids, Vol. 2, edited by J. F. Johnson and R. S. Porter (Plenum, New York, 1974), pp. 383–401.
[CrossRef]

Mazkedian, S.

S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
[CrossRef]

Melone, S.

S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
[CrossRef]

Neville, A. C.

A. C. Neville and S. Caveney, “Scarabaeid Beetle Exocuticle as an Optical Analogue of Cholesteric Liquid Crystals,” Biol. Rev. 44, 531–562, (1969).
[CrossRef] [PubMed]

Nityanda, R.

R. Nityanda, “On the Theory of Light Propagation in Cholesteric Liquid Crystals,” Mol. Cryst. Liq. Cryst. 21, 315–331, (1973).
[CrossRef]

Rao, K. N. S.

S. Chandrasekhar and K. N. S. Rao “Optical Rotatory Power of Liquid Crystals,” Acta. Crystallogr. A 24, 445–451, (1968).
[CrossRef]

Rustichelli, F.

S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
[CrossRef]

Wax, S. G.

A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
[CrossRef]

A.I.Ch.E. (1)

A. A. Kozinski, G. J. Kizior, and S. G. Wax, “Separations with Protein Liquid Crystals,” A.I.Ch.E. 20, 6, 1104–1109, (1974).
[CrossRef]

Acta. Crystallogr. A (1)

S. Chandrasekhar and K. N. S. Rao “Optical Rotatory Power of Liquid Crystals,” Acta. Crystallogr. A 24, 445–451, (1968).
[CrossRef]

Biol. Rev. (1)

A. C. Neville and S. Caveney, “Scarabaeid Beetle Exocuticle as an Optical Analogue of Cholesteric Liquid Crystals,” Biol. Rev. 44, 531–562, (1969).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (4)

R. C. Jones, wrote a series of eight noteworthy papers on optics, the last of which is in J. Opt. Soc. Am. 46, 126, (1956).
[CrossRef]

G. H. Conners, “Electromagnetic wave propagation in cholesteric materials,” J. Opt. Soc. Am. 58, 875–879, (1968).
[CrossRef]

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 4, 502–510 (1972).
[CrossRef]

D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: Application to liquid-crystal twist cells,” J. Opt. Soc. Am. 61, 11, 1374–1380, (1973).

J. Physique (1)

S. Mazkedian, S. Melone, and F. Rustichelli, “On the Circular Dichroism and Rotatory Dispersion in Cholesteric Liquid Crystals with a Pitch Gradient,” J. Physique 37, 731–736, (1976).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

R. Nityanda, “On the Theory of Light Propagation in Cholesteric Liquid Crystals,” Mol. Cryst. Liq. Cryst. 21, 315–331, (1973).
[CrossRef]

Other (3)

A. C. Eringen and J. D. Lee, Liquid Crystals and Ordered Fluids, Vol. 2, edited by J. F. Johnson and R. S. Porter (Plenum, New York, 1974), pp. 383–401.
[CrossRef]

L. E. Hajdo and A. C. Eringen, “A Theory of Light Reflection by Cholesteric Liquid Crystals Possessing a Tilted Structure,” (unpublished).

L. M. Brekhovskikh, “Waves in Layered Media” (Academic, New York, 1960).

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Figures (6)

FIG. 1
FIG. 1

Multilayer model of cholesteric liquid crystal slab.

FIG. 2
FIG. 2

Left- and right-hand circularly polarized reflections off a cholesteric liquid crystal slab possessing a pitch gradient; = 2.25; δ =.033; △, left; □ right; - - - - no pitch gradient.

FIG. 3
FIG. 3

Single-layer model of a cholesteric liquid crystal slab.

FIG. 4
FIG. 4

Reflection off a cholesteric liquid crystal slab with a continuously varying pitch. Input is a unit left-polarized wave; = 1.0; δ =.05; σ = 0.002; —, right-polarized reflection; —, left-polarized reflection.

FIG. 5
FIG. 5

Reflection off a cholesteric liquid crystal slab with a continuously varying pitch. Input is a unit left-polarized wave; = 2.25; δ =.003; σ =.002; —, right-polarized reflection; —, left-polarized reflection.

FIG. 6
FIG. 6

Reflection off a cholesteric liquid crystal slab with a continuously varying pitch. Input is a unit left-polarized wave. = 2.25; δ =.067; σ =.002; —, right-polarized reflection; —, left-polarized reflection.

Equations (51)

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θ 1 = α 1 z
θ 2 = ( z + l ) α 2 α 1 l
θ m = α m [ z + ( m 1 ) l ] l t = 1 m 1 α t ,
θ m = α m z + m l α m l t = 1 m α t ,
θ m α m z + ϕ m .
U 1 + k 2 ( U 1 + δ e 2 i α z U 2 ) = 0 , U 2 + k 2 ( U 2 + δ e 2 i α z U 1 ) = 0 .
δ ( 11 22 ) / 2 ,
( 11 + 22 ) / 2 ,
k 2 ω 2 / c 2 ,
U m 1 + k 2 ( U m 1 + δ e 2 i ϕ m e 2 i α m z U m 2 ) = 0 , U m 2 + k 2 ( U m 2 + δ e 2 i ϕ m e 2 i α m z U m 1 ) = 0 .
U m 1 4 i α m U m 1 + ( 2 k 2 4 α m 2 ) U m 1 4 i α m k 2 U m 1 + [ k 4 ( 1 δ 2 ) 4 α m 2 k 2 ] U m 1 = 0 .
U m 1 = e λ m z
U m 1 = t = 1 4 A m t e λ m t z .
U m 2 = t = 1 4 B m t e ( λ m t 2 i α m ) z ,
B m t = ( e 2 i ϕ m / δ k 2 ) A m t ( λ m t 2 + k 2 ) .
U 1 + = T 1 e iKz , U 2 + = T 2 e iKz .
U 1 = R 1 e iKz + I 1 e iKz , U 2 = R 2 e iKz + I 2 e iKz .
U 1 + k 2 ( U 1 + δ e 2 i γ z U 2 ) = 0 , U 2 + k 2 ( U 2 + δ e 2 i γ z U 1 ) = 0 ,
γ = γ ( z ) .
γ = α ( 1 + σ z ) ,
α 2 π / p ,
U 1 + k 2 U 1 = k 2 δ e 2 i γ z U 2 , U 2 + k 2 U 2 = k 2 δ e 2 i γ z U 1 .
U 1 ~ e i γ z , U 2 ~ e i γ z .
U 4 i α U + ( 2 k 2 4 α 2 ) U 4 i α k 2 U + [ k 4 ( 1 δ 2 ) 4 α 2 k 2 ] U = 0 .
U = U 0 e ( i α + h ) z ,
h = h ( α , k , δ ) .
h 2 = ( α 2 + k 2 ) ± 2 α k ( 1 + k 2 δ 2 / 4 α 2 ) 1 / 2 .
h 2 k 3 δ 2 / 4 α ( α k ) 2 .
h m 2 k 3 δ 2 / 4 α m ( α m k ) 2 ,
e ± h m Δ z
e ± h m + 1 Δ z .
e h 1 Δ z e h 2 Δ z e h n Δ z = exp ( t = 1 n h t Δ z ) .
h 2 ( z ) = k 3 δ 2 / 4 γ ( z ) [ γ ( z ) k ] 2 .
J ( z ) exp [ I ( z ) ] ,
I ( z ) z h ( t ) d t
γ k = ± ( k 3 δ 2 / 4 γ ) 1 / 2 .
γ k ( k 2 δ 2 / 4 ) 1 / 2 ,
γ k ± k δ / 2 .
h ( z ) ± [ k 2 δ 2 / 4 ( γ k ) 2 ] 1 / 2 .
e x + = e x e = e x ( 1 + + 2 / 2 ! )
I ( z ) = 1 α σ { ( γ k ) 2 [ a 2 ( γ k ) 2 ] 1 / 2 + a 2 2 sin 1 ( γ k a ) } | s z
a k δ / 2
lim σ 0 I ( z ) = k δ z / 2 .
U 1 = ( i δ / 4 ) A exp { [ i + ( δ / 2 ) ] k z } , U 2 = A exp { [ i ( δ / 2 ) ] k z } ;
U 1 = B e ikz , U 2 = ( δ B / 8 ) e 3 ikz ;
U 1 = ( δ C / 8 ) e 3 ikz , U 2 = C e ikz ;
U 1 = D exp { [ i ( δ / 2 ) ] k z } , U 2 = [ i ( δ / 4 ) ] D exp { [ i + ( δ / 2 ) ] k z } .
U 1 = ( i δ / 4 ) A exp [ i γ z + I ( z ) ] , U 2 = A exp [ i γ z + I ( z ) ] ;
U 1 = B exp ( ikz ) , U 2 = ( δ B / 8 ) exp ( 3 ikz ) ;
U 1 = ( δ C / 8 ) exp ( 3 ikz ) , U 2 = C exp ( ikz ) ;
U 1 = D exp [ i γ z I ( z ) ] , U 2 = ( i δ / 4 ) D exp [ i γ z I ( z ) ] .