Abstract

Light propagation in uniaxial chiral media with large pitch is studied. In these systems there are forbidden zones for extraordinary beams, which lead to effective reflection on zone boundaries and to wave damping inside the forbidden zone. We analyze the vicinities of the turning points and the transition of an extraordinary wave through the forbidden zone. Narrow forbidden zones with merging turning points are studied in detail. The transition through the forbidden zone is studied experimentally in nematic liquid crystal doped with a chiral addition. There is a good agreement between experimental results and theoretical calculations.

© 2008 Optical Society of America

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References

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  1. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).
  2. S. Chandrasekhar, H.-S. Kitzerow, and C. Bahr, Chirality in Liquid Crystals (Springer-Verlag, 2001).
  3. S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, 1977).
  4. D. W. Berreman and T. J. Scheffer, "Order versus temperature in cholesteric liquid crystals from reflectance spectra," Phys. Rev. A 5, 1397-1403 (1971).
    [CrossRef]
  5. D. W. Berreman, "Optics in stratified and anisotropic media: 4×⁢⁢4-matrix formulation," J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  6. D. W. Berreman, "Stratified media: optics in smoothly varying anisotropic planar structures: application to liquid-crystal twist cells," J. Opt. Soc. Am. 63, 1374-1379 (1973).
    [CrossRef]
  7. S. P. Palto, "An algorithm for solving the optical problem for stratified anisotropic media," JETP 92, 552-560 (2001).
    [CrossRef]
  8. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993).
  9. E. I. Kats, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004-1009 (1970).
  10. V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, 1992).
    [CrossRef]
  11. C. Oldano and S. Ponti, "Acoustic wave propagation in structurally helical media," Phys. Rev. E 63, 011703 (2000).
    [CrossRef]
  12. S. Ponti, C. Oldano, and M. Becchi, "Bloch wave approach to the optics of crystals," Phys. Rev. E 64, 021704 (2001).
    [CrossRef]
  13. V. S. Liberman and B. Ya. Zel'dovich, "Birefringence by a smoothly inhomogeneous locally isotropic medium," Phys. Rev. E 49, 2389-2396 (1994).
    [CrossRef]
  14. A. Yu. Savchenko and B. Ya. Zel'dovich, "Birefringence by asmoothly in homogeneous locally isotropic medium: three-dimensional case," Phys. Rev. E 50, 2287-2292 (1994).
    [CrossRef]
  15. E. V. Aksenova, A. A. Karetnikov, A. P. Kovshik, V. P. Romanov, and A. Yu. Val'kov, "Return back of the extraordinary beam for oblique incidence in helical liquid crystals with large pitch," Europhys. Lett. 69, 68-74 (2005).
    [CrossRef]
  16. N. Fröman and P. O. Fröman, JWKB Approximation (North-Holland, 1965).
  17. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevsky, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1995).

2005 (1)

E. V. Aksenova, A. A. Karetnikov, A. P. Kovshik, V. P. Romanov, and A. Yu. Val'kov, "Return back of the extraordinary beam for oblique incidence in helical liquid crystals with large pitch," Europhys. Lett. 69, 68-74 (2005).
[CrossRef]

2001 (2)

S. P. Palto, "An algorithm for solving the optical problem for stratified anisotropic media," JETP 92, 552-560 (2001).
[CrossRef]

S. Ponti, C. Oldano, and M. Becchi, "Bloch wave approach to the optics of crystals," Phys. Rev. E 64, 021704 (2001).
[CrossRef]

2000 (1)

C. Oldano and S. Ponti, "Acoustic wave propagation in structurally helical media," Phys. Rev. E 63, 011703 (2000).
[CrossRef]

1994 (2)

V. S. Liberman and B. Ya. Zel'dovich, "Birefringence by a smoothly inhomogeneous locally isotropic medium," Phys. Rev. E 49, 2389-2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel'dovich, "Birefringence by asmoothly in homogeneous locally isotropic medium: three-dimensional case," Phys. Rev. E 50, 2287-2292 (1994).
[CrossRef]

1973 (1)

1972 (1)

1971 (1)

D. W. Berreman and T. J. Scheffer, "Order versus temperature in cholesteric liquid crystals from reflectance spectra," Phys. Rev. A 5, 1397-1403 (1971).
[CrossRef]

1970 (1)

E. I. Kats, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004-1009 (1970).

Europhys. Lett. (1)

E. V. Aksenova, A. A. Karetnikov, A. P. Kovshik, V. P. Romanov, and A. Yu. Val'kov, "Return back of the extraordinary beam for oblique incidence in helical liquid crystals with large pitch," Europhys. Lett. 69, 68-74 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

JETP (1)

S. P. Palto, "An algorithm for solving the optical problem for stratified anisotropic media," JETP 92, 552-560 (2001).
[CrossRef]

Phys. Rev. A (1)

D. W. Berreman and T. J. Scheffer, "Order versus temperature in cholesteric liquid crystals from reflectance spectra," Phys. Rev. A 5, 1397-1403 (1971).
[CrossRef]

Phys. Rev. E (4)

C. Oldano and S. Ponti, "Acoustic wave propagation in structurally helical media," Phys. Rev. E 63, 011703 (2000).
[CrossRef]

S. Ponti, C. Oldano, and M. Becchi, "Bloch wave approach to the optics of crystals," Phys. Rev. E 64, 021704 (2001).
[CrossRef]

V. S. Liberman and B. Ya. Zel'dovich, "Birefringence by a smoothly inhomogeneous locally isotropic medium," Phys. Rev. E 49, 2389-2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel'dovich, "Birefringence by asmoothly in homogeneous locally isotropic medium: three-dimensional case," Phys. Rev. E 50, 2287-2292 (1994).
[CrossRef]

Sov. Phys. JETP (1)

E. I. Kats, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004-1009 (1970).

Other (7)

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, 1992).
[CrossRef]

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999).

S. Chandrasekhar, H.-S. Kitzerow, and C. Bahr, Chirality in Liquid Crystals (Springer-Verlag, 2001).

S. Chandrasekhar, Liquid Crystals (Cambridge U. Press, 1977).

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993).

N. Fröman and P. O. Fröman, JWKB Approximation (North-Holland, 1965).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevsky, Electrodynamics of Continuous Media (Butterworth-Heinemann, 1995).

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Figures (6)

Fig. 1
Fig. 1

Cross section of the wave vector surface for ε a > 0 . The cross section is made for ξ = 0 . Varying ξ does not change the surface of the ordinary beam (sphere) but does change the surface of the extraordinary beam (ellipsoid), since the length of the semiaxis varies. In particular, changing ξ by π 2 transforms the ellipsoid into a sphere. The number of waves propagating in the medium depends on the value of k : vectors 1 and 2 correspond to the case of two propagating waves, and vector 3 to the case of one propagating wave.

Fig. 2
Fig. 2

Projection of the extraordinary beam trajectory in the plane x z and the trajectory of the ordinary beam in the chiral medium for various angles of incidence: (a) transmitted ordinary, 1, and extraordinary, 2, beams for ε a > 0 ; (b) forbidden zone for the extraordinary beam (cross-hatched area).

Fig. 3
Fig. 3

Angular dependance of the reflection V 2 (curves 1, 2, 3) and transmission W 2 (curves 1 , 2 , 3 ) coefficients in the chiral media with ε = 2.863 , ε = 2.291 for different values of Ω = p 0 λ : 1 Ω = 20 , 2 Ω = 50 , 3 Ω = 200 ; χ 2 is the angle between the beam and the z axis on the plane ξ = π 2 . The critical beam corresponds to χ 2 = arcsin ( ε ε ) 1 2 = 63.46 ° . The hatched area shows the presence of the forbidden zone. All the curves intersect at point C, corresponding to the critical beams.

Fig. 4
Fig. 4

Schematic for measurements of transmitted beams. He–Ne, laser; P 1 , P 2 , polarizers; λ 2 , half-wave plate; Ph, photodetector; V, digital voltmeter; G, goniometer; C, liquid crystal cell.

Fig. 5
Fig. 5

Cell with twisted liquid crystal and schematic beam trajectories: 1, twisted liquid crystal; 2, 3 prism glasses with height 12 mm, larger base 37 mm and base angle γ 0 = 70 ° . Angles χ 1 and χ 2 are the angles of incidence and refraction on the liquid crystal cell.

Fig. 6
Fig. 6

Transmission coefficient measured for the ordinary (open circles) and extraordinary (filled circles) beams. The solid curves are calculations: 1, Fresnel’s formulae for the ordinary beam (52); 2, equation (51) for the extraordinary beam. (a) Wide range of angles, (b) narrow range of angles in the vicinity of the critical beam.

Equations (71)

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n 0 ( z ) = ( cos ( q 0 z + ϕ 0 ) , sin ( q 0 z + ϕ 0 ) , 0 ) ,
ε l k ( z ) = ε δ l k + ε a n l 0 ( z ) n k 0 ( z ) , l , k = 1 , 2 , 3 .
curl E ( r ) = i k 0 μ ̂ H ( r ) , curl H ( r ) = i k 0 ε ̂ ( z ) E ( r ) ,
F ( k , z ) = d r F ( r ) e i k r .
ξ ( E x E y H x H y ) = i Ω ( 0 0 0 α 0 0 1 0 ε x y ε y y + ε ( α 1 ) 0 0 ε x x ε x y 0 0 ) ( E x E y H x H y ) ,
ξ Φ ( ξ ) = i Ω A ̂ ( ξ ) Φ ( ξ ) .
E ± ( j ) ( r ) = E 0 ( j ) A ( j ) ( k ; z , z 0 ) e ( j ) ± ( k , z ) exp [ i k r ± i z 0 z k z ( j ) ( k , z ) dz ] ,
k z ( o ) ( k , z ) k z ( o ) ( k ) = ε k 0 2 k 2 = k 0 ε α ,
k z ( e ) ( k , z ) = [ ε k 0 2 k 2 ε a ε ( k n 0 ( z ) ) 2 ] 1 2 = k 0 [ α ε + ( 1 α ) ε a sin 2 ξ ] 1 2 ,
A ( o ) ( k ; z , z 0 ) = 1 , A ( e ) ( k ; z , z 0 ) = [ ε 2 k 0 2 + ε a ( k n 0 ( z ) ) 2 ε 2 k 0 2 + ε a ( k n 0 ( z 0 ) ) 2 ] 1 2 [ k z ( e ) ( k , z 0 ) k z ( e ) ( k , z ) ] 1 2 ,
e ( o ) ± ( k , z ) = 1 ( k 0 2 ε k 2 cos 2 ξ ) 1 2 ( ± k z ( o ) sin ξ , k z ( o ) cos ξ , k sin ξ ) ,
e ( e ) ± ( k , z ) = ε [ ( k 0 2 ε k 2 cos 2 ξ ) ( k 0 2 ε 2 + ε a k 2 cos 2 ξ ) ] 1 2 ( ( k 0 2 ε k 2 ) cos ξ , k 0 2 ε sin ξ , k k z ( e ) cos ξ ) .
n ( o ) = ε , n ( e ) = ( ε ε ε cos 2 θ + ε sin 2 θ ) 1 2 ,
sin 2 ξ α 1 α ε ε a .
sin 2 ξ * = α 1 α ε ε a .
sin 2 ξ * * = α ( 1 α ) .
ξ ξ * Ω 2 3 ,
ξ ξ * * Ω 1 2 .
i ε ξ E z + Ω ε x x C 0 + Ω ε x y E y = 0 ,
ξ 2 E y + Ω 2 ε x y C 0 + Ω 2 ε y y E y = 0.
E z = i Ω ε [ C 0 ξ ( 1 + ε ε a ) + ε ε a 4 C 0 sin 2 ξ + ε ε a E y cos ξ sin ξ d ξ ] + C .
ξ E z + i Ω ε a ε sin ξ cos ξ E y = 0 , ξ 2 E y + Ω 2 ε a sin 2 ξ E y = 0 .
E = E ( e ) + + E ( e ) + E ( o ) ,
E ( o ) = ( 0 , 0 , E ( o ) ) ,
E ( e ) ± = C ( e ) ± ε a 1 4 sin ξ 1 2 exp ( ± i Ω ε a 1 2 0 ξ sin ξ d ξ ) × ( 0 , 1 , ε a 1 2 ε 1 2 sign ( sin ξ ) cos ξ ) .
ζ E z + i ε a 1 2 ε 1 2 ζ E y = 0 , ζ 2 E y + ζ 2 E y = 0 ,
E y , > = ζ 1 2 [ A > H 1 4 ( 1 ) ( 1 2 ζ 2 ) + B > H 1 4 ( 2 ) ( 1 2 ζ 2 ) ]
E y , < = ζ 1 2 [ A < H 1 4 ( 1 ) ( 1 2 ζ 2 ) + B < H 1 4 ( 2 ) ( 1 2 ζ 2 ) ]
E y , > 2 i Γ ( 3 4 ) ( A > + B > ) + 2 Γ ( 3 4 ) ζ π [ ( 1 + i ) A > + ( 1 i ) B > ] ,
E y , < 2 i Γ ( 3 4 ) ( A < + B < ) 2 Γ ( 3 4 ) ζ π [ ( 1 + i ) A < + ( 1 i ) B < ] .
A > = 1 2 ( B < e i 3 π 4 + B > e i π 4 )
A < = 1 2 ( B < e i π 4 + B > e i 3 π 4 ) .
H ν ( j ) ( u ) = 2 π u exp [ ± i ( u π 2 ν π 4 ) ] [ 1 + O ( u 1 ) ] ,
E y , > = C > ( e ) + 1 ε a 1 4 ξ exp ( i Ω ε a 1 2 ξ 2 2 ) + C > ( e ) 1 ε a 1 4 ξ exp ( i Ω ε a 1 2 ξ 2 2 ) ,
E y , < = C < ( e ) + 1 ε a 1 4 ξ exp ( i Ω ε a 1 2 ξ 2 2 ) + C < ( e ) 1 ε a 1 4 ξ exp ( i Ω ε a 1 2 ξ 2 2 ) ,
E y , > = 2 π ζ { A > exp [ i ( ζ 2 2 3 π 8 ) ] + B > exp [ i ( ζ 2 2 3 π 8 ) ] } , )
E y , < = 2 π ζ { A < exp [ i ( ζ 2 2 3 π 8 ) ] + B < exp [ i ( ζ 2 2 3 π 8 ) ] } .
A > = π 1 2 2 Ω 1 4 ε a 1 8 e 3 π i 8 C > ( e ) + ,
B > = π 1 2 2 Ω 1 4 ε a 1 8 e 3 π i 8 C > ( e ) ,
A < = π 1 2 2 Ω 1 4 ε a 1 8 e 3 π i 8 C < ( e ) ,
B < = π 1 2 2 Ω 1 4 ε a 1 8 e 3 π i 8 C < ( e ) + .
C > ( e ) + = 1 2 ( C < ( e ) + i C > ( e ) )
C < ( e ) = 1 2 ( i C < ( e ) + + C > ( e ) ) .
C > ( e ) + 2 + C < ( e ) 2 = C < ( e ) + 2 + C > ( e ) 2 .
E z , > = F z i π 1 2 2 3 2 Γ ( 3 4 ) ε a 1 2 ε 1 2 ζ 2 { A > [ H 1 4 ( 1 ) ( ζ 2 2 ) H 3 4 ( ζ 2 2 ) H 3 4 ( 1 ) ( ζ 2 2 ) H 1 4 ( ζ 2 2 ) ] + B > [ H 1 4 ( 2 ) ( ζ 2 2 ) H 3 4 ( ζ 2 2 ) H 3 4 ( 2 ) ( ζ 2 2 ) H 1 4 ( ζ 2 2 ) ] } ,
ξ 2 E x + α Ω 2 ε x x E x + α Ω 2 ε x y E y = 0 , ξ 2 E y + Ω 2 ( ε y y k 2 k 0 2 ) E y + Ω 2 ε x y E x = 0 .
α = δ α Ω 1 , 0 < δ α 1 ,
δ α Ω ε a Δ 2 q 0 2 4 ε .
Δ = 2 q 0 arc cos ( cot χ 0 ε ε a ) .
ξ Ω 1 2 .
z 2 E + k z ( e ) 2 E = 0
ξ 2 E + Ω 2 [ α ε + ( 1 α ) ε a sin 2 ξ ] E = 0 .
ζ 2 E + [ ζ 2 δ α ε ε a 1 2 + Ω 1 ( δ α ζ 2 ζ 4 3 ε a 1 2 ) + ] E = 0 .
E ( ζ ) = E 0 ( ζ ) + Ω 1 E 1 ( ζ ) + Ω 2 E 2 ( ζ ) + .
ζ 2 E 0 + ( ζ 2 ψ 2 ) E 0 = 0 ,
ψ = [ δ α ( ε / ε a 1 2 ) ] 1 2 .
E 0 = A 1 D 1 2 i ψ 2 2 ( 2 e i π 4 ζ ) + A 2 D 1 2 i ψ 2 2 ( 2 e i 3 π 4 ζ ) .
D ν ( u ) = e u 2 4 u ν [ 1 + O ( u 2 ) ] , arg u ( π 2 , π 2 ) ,
D ν ( u ) = [ e u 2 4 u ν 2 π Γ 1 ( ν ) e u 2 4 ± i π ν u 1 ν ] [ 1 + O ( u 2 ) ] ,
E 0 , > = G 3 ζ 1 2 exp [ i ψ 2 2 ( ζ 2 ψ 2 ln 2 ζ ψ ) ]
E 0 , < = ζ 1 2 { G 2 exp [ i ψ 2 2 ( ζ 2 ψ 2 ln 2 ζ ψ ) ] + G 1 exp [ i ψ 2 2 ( ζ 2 ψ 2 ln 2 ζ ψ ) ] }
G 1 = A 1 2 π Γ 1 ( 1 2 + i ψ 2 2 ) exp [ i ( π 8 ψ 2 4 ln 2 + ψ 2 2 ln ψ ) 1 4 ln 2 + π ψ 2 8 ] ,
G 2 = A 1 exp [ i ( 3 π 8 ψ 2 4 ln 2 + ψ 2 2 ln ψ ) 1 4 ln 2 + 3 π ψ 2 8 ] ,
G 3 = A 1 exp [ i ( π 8 + ψ 2 4 ln 2 ψ 2 2 ln ψ ) 1 4 ln 2 π ψ 2 8 ] .
V = [ 1 + exp ( π ψ 2 ) ] 1 2 exp [ i ( π 2 + ϕ ) ] ,
W = [ 1 + exp ( π ψ 2 ) ] 1 2 exp ( i ϕ ) ,
ϕ = arg Γ ( 1 2 + i ψ 2 2 ) ψ 2 ln 2 2 + ψ 2 ln ψ .
V 2 = ( 1 + e π ψ 2 ) 1 , W 2 = ( 1 + e π ψ 2 ) 1 .
n p sin χ 1 = ε sin χ 2 .
n p sin χ 1 = ε sin χ 2 .
I ( χ 1 ) = I 0 [ 1 tan ( χ 2 χ 1 ) tan ( χ 2 + χ 1 ) ] , χ 2 = arcsin n p sin χ 1 ε .

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