Abstract

We discuss a planar nematic liquid-crystal structure with the director periodically twisted in the light propagation direction (periodically twisted nematic) as a medium, which exhibits bandgap and strong reflection for any polarization at normal incidence. This is in contrast to the cholesteric liquid crystals, which reflect only one of the two circular polarizations. The size of the bandgap depends on the modulation profile and amplitude, and its maximal magnitude is smaller, but of the same order as the bandgap of cholesterics with similar material parameters (refractive indices and pitch). The second-order bandgap reflection exhibits polarization universality only for large enough modulation.

© 2006 Optical Society of America

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  1. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993).
  2. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).
  3. V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, 1992).
    [CrossRef]
  4. V. A. Belyakov and V. E. Dmitrienko, "The blue phase of liquid crystals," Sov. Phys. Usp. 146, 369-415 (1985).
    [CrossRef]
  5. E. I. Katz, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004 (1971).
  6. R. S. Akopyan, B. Ya. Zel'dovich, and N. V. Tabiryan, "Optics of a chiral liquid crystal far from a Bragg resonance," Sov. Phys. JETP 56, 1024-1027 (1982).
  7. N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zeldovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
    [CrossRef]
  8. N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zeldovich, "Polarization-universal bandgap in periodically twisted nematics," Opt. Lett. 31, 1678-1680 (2006).
    [CrossRef] [PubMed]
  9. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  10. B. Ya. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear OpticsCRC Press, 1995.
  11. N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zeldovich, "High-efficiency energy transfer due to stimulated orientational scattering of light in nematic liquid crystals," J. Opt. Soc. Am. B 18, 1203-1205 (2001).
    [CrossRef]
  12. H. Sarkissian, C. C. Tsai, B. Ya. Zeldovich, and N. V. Tabiryan, "Beam combining using orientational stimulated scattering in liquid crystals," J. Opt. Soc. Am. B 22, 2628-2634 (2005).
    [CrossRef]
  13. N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
    [CrossRef] [PubMed]
  14. I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

2006

2005

H. Sarkissian, C. C. Tsai, B. Ya. Zeldovich, and N. V. Tabiryan, "Beam combining using orientational stimulated scattering in liquid crystals," J. Opt. Soc. Am. B 22, 2628-2634 (2005).
[CrossRef]

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

2004

N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
[CrossRef] [PubMed]

2001

1986

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zeldovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

1985

V. A. Belyakov and V. E. Dmitrienko, "The blue phase of liquid crystals," Sov. Phys. Usp. 146, 369-415 (1985).
[CrossRef]

1982

R. S. Akopyan, B. Ya. Zel'dovich, and N. V. Tabiryan, "Optics of a chiral liquid crystal far from a Bragg resonance," Sov. Phys. JETP 56, 1024-1027 (1982).

1971

E. I. Katz, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004 (1971).

1969

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Akopyan, R. S.

R. S. Akopyan, B. Ya. Zel'dovich, and N. V. Tabiryan, "Optics of a chiral liquid crystal far from a Bragg resonance," Sov. Phys. JETP 56, 1024-1027 (1982).

Belyakov, V. A.

V. A. Belyakov and V. E. Dmitrienko, "The blue phase of liquid crystals," Sov. Phys. Usp. 146, 369-415 (1985).
[CrossRef]

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

de Gennes, P. G.

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

Dmitrienko, V. E.

V. A. Belyakov and V. E. Dmitrienko, "The blue phase of liquid crystals," Sov. Phys. Usp. 146, 369-415 (1985).
[CrossRef]

Guizar-Iturbide, I.

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Hrozhyk, U.

N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
[CrossRef] [PubMed]

Katz, E. I.

E. I. Katz, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004 (1971).

Khoo, C.

C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Lazo-Martinez, I.

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Mamaev, A. V.

B. Ya. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear OpticsCRC Press, 1995.

Olivares-Perez, A.

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Prost, J.

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

Ramos-Garcia, R.

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Sarkissian, H.

Serak, S.

N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
[CrossRef] [PubMed]

Shkunov, V. V.

B. Ya. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear OpticsCRC Press, 1995.

Sukhov, A. V.

Tabiryan, N.

N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
[CrossRef] [PubMed]

Tabiryan, N. V.

Trevino-Palacios, C. G.

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Tsai, C. C.

Wu, S. T.

C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).

Zeldovich, B. Ya.

Zel'dovich, B. Ya.

R. S. Akopyan, B. Ya. Zel'dovich, and N. V. Tabiryan, "Optics of a chiral liquid crystal far from a Bragg resonance," Sov. Phys. JETP 56, 1024-1027 (1982).

Bell Syst. Tech. J.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

N. V. Tabiryan, A. V. Sukhov, and B. Ya. Zeldovich, "The orientational optical nonlinearity of liquid crystals," Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

N. Tabiryan, U. Hrozhyk, and S. Serak, "Nonlinear refraction in photoinduced isotropic state of liquid crystalline azobenzenes," Phys. Rev. Lett. 93, 113901 (2004).
[CrossRef] [PubMed]

Proc. SPIE

I. Lazo-Martinez, I. Guizar-Iturbide, A. Olivares-Perez, C. G. Trevino-Palacios, and R. Ramos-Garcia, "Colossal optical nonlinear effects at the nematic-isotropic temperature transition in azo-dye-doped liquid crystals," in Liquid Crystals IX, I.-C. Khoo, ed., Proc. SPIE 5936, 201-208 (2005).

Sov. Phys. JETP

E. I. Katz, "Optical properties of cholesteric liquid crystals," Sov. Phys. JETP 32, 1004 (1971).

R. S. Akopyan, B. Ya. Zel'dovich, and N. V. Tabiryan, "Optics of a chiral liquid crystal far from a Bragg resonance," Sov. Phys. JETP 56, 1024-1027 (1982).

Sov. Phys. Usp.

V. A. Belyakov and V. E. Dmitrienko, "The blue phase of liquid crystals," Sov. Phys. Usp. 146, 369-415 (1985).
[CrossRef]

Other

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

C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993).

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

B. Ya. Zeldovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear OpticsCRC Press, 1995.

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Figures (4)

Fig. 1
Fig. 1

Schematic structure of the cell with PTNs for a particular example of sinusoidal modulation [Eq. (5)]. The nematic director is modulated in the direction of light propagation. Such modulation results in a polarization-independent lowest-order bandgap.

Fig. 2
Fig. 2

Intensity reflection coefficient of a PTN cell with thickness L = 35 μ m , n c = 1.65 , n o = 1.5 , sinusoidal modulation amplitude θ s = 0.2 rad , λ vac = 1.06 μ m , R ( ω = ω 0 ) = 99 % , i.e., κ s 1 L = 3 . The bandgap is characterized by the first zeros of the reflection coefficient.

Fig. 3
Fig. 3

Dispersion curves showing the relation between the quasi-momentum k and the frequency of incident light ω for a sinusoidally modulated PTN structure with n e = 1.7 , n o = 1.5 , q = 1.6 × 10 7 1 m , and θ 0 = 0.8 rad . The bandgap at ω = 1.5 × 10 15 s 1 exists for both polarizations.

Fig. 4
Fig. 4

Second-order bandgaps in the PTN structure with sinusoidal modulation and with n e = 1.7 , n o = 1.5 , q = 1.6 × 10 7 1 m , as observed around ω = 3 × 10 15 s 1 . (a) For θ s = 0.7 rad the two second-order bandgaps are separated by a distance larger than their widths. (b) When θ s = π 2 rad , the averaged refractive indices seen by the two propagating modes are close enough to each other so that the bandgaps overlap. This overlap, and hence polarization universality, exists when θ s is larger than 0.92 rad .

Equations (45)

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n ( z ) = e x cos [ θ ( z ) ] + e y sin [ θ ( z ) ] ,
θ ( z ) = q Ch z = 2 π z Λ Ch = π z Λ phys .
λ vac ( center , ChLC ) = Λ Ch ( n e + n o ) 2 Λ phys ( n e + n o ) ,
Δ ω Ch ω Δ λ Ch λ ( n e n o ) [ ( n e + n o ) 2 ] .
θ ( z ) = θ s sin ( q PTN z ) ( sinusoidal modulation )
θ ( z ) = θ r sin ( q PTN z ) sin ( q PTN z ) ( rectangular modulation ) ,
ϵ i j = ϵ [ ϵ δ i j + ( ϵ ϵ ) n i n j ] ϵ [ n o 2 δ i j + ( n e 2 n o 2 ) n i n j ] ,
d 2 E i d z 2 + ω 2 c 2 ϵ i j ϵ vac E j = 0 .
d 2 E x d z 2 + ω 2 c 2 [ n e 2 + n o 2 2 E x + n e 2 n o 2 2 cos [ 2 θ ( z ) ] E x + n e 2 n o 2 2 sin [ 2 θ ( z ) ] E y ] = 0 ,
d 2 E y d z 2 + ω 2 c 2 [ n e 2 + n o 2 2 E y n e 2 n o 2 2 cos [ 2 θ ( z ) ] E y + n e 2 n o 2 2 sin [ 2 θ ( z ) ] E x ] = 0 .
cos [ 2 θ ( z ) ] = m = c m exp ( i m q z ) ,
sin [ 2 θ ( z ) ] = m = s m exp ( i m q z ) ,
c 2 p + 1 = 0 , c 2 p = J 2 p ( 2 θ s ) , s 2 p + 1 = i J 2 p + 1 ( 2 θ s ) ,
s 2 p = 0 , p = 0 , ± 1 , ± 2 ,
d 2 E x d z 2 + k e 2 E x + ω 2 c 2 n e 2 n o 2 2 m 0 ( c m E x + s m E y ) exp ( i m q z ) = 0 ,
d 2 E y d z 2 + k o 2 E y + ω 2 c 2 n e 2 n o 2 2 m 0 ( s m E x c m E y ) exp ( i m q z ) = 0 ,
k e = ω c 1 2 [ n e 2 + n o 2 + c 0 ( n e 2 n o 2 ) ] ,
k o = ω c 1 2 [ n e 2 + n o 2 c 0 ( n e 2 n o 2 ) ] .
E x ( z ) = ( k o k e ) 1 4 A ( z ) exp [ i ( k e γ ) z ] ,
E y ( z ) = ( k o k e ) 1 4 B ( z ) exp [ i ( k o γ ) z ] ,
γ = 0.5 ( k e + k o m q ) ,
d A d z = i γ A + i κ B , d B d z = i γ B i κ * A ,
κ = 0.25 s m ( n e 2 n o 2 ) ( ω c ) 2 k e k o .
Δ ω ( o e ) = ω 0 s m ( n e n o ) n , n = ( n e + n o ) 2 .
r = i κ * sinh ( μ L ) μ cosh ( μ L ) i γ sinh ( μ L ) .
E x ( z ) = A ( z ) exp [ i ( k e γ ) z ] + B ( z ) exp [ i ( k e γ ) z ] ,
E y ( z ) 0 ,
Δ ω ( e e ) = ω 0 c m ( n e n o ) [ ( n e + n o ) 2 ] .
E y ( z ) = A ( z ) exp [ i ( k o γ ) z ] + B ( z ) exp [ i ( k o γ ) z ] ,
E x ( z ) 0 ,
E x ( z ) = A x ( z ) exp [ i ( q m 2 ) z ] + B x ( z ) exp [ i ( q m 2 ) z ] ,
E y ( z ) = A y ( z ) exp [ i ( q m 2 ) z ] + B y ( z ) exp [ i ( q m 2 ) z ] .
d d z [ A x A y B x B y ] = M ̂ [ A x A y B x B y ] ,
M ̂ = [ i γ e 0 i κ c i κ s 0 i γ o i κ s i κ c i ( κ c ) * i ( κ s ) * i γ e 0 i ( κ s ) * i ( κ c ) * 0 i γ o ] ,
γ e = k e q m 2 , γ o = k o q m 2 ,
κ c = 0.5 c m ( ω c ) ( n e n o ) , κ s = 0.5 s m ( ω c ) ( n e n o ) ,
μ 4 + { γ o 2 + γ e 2 2 κ s 2 2 κ c 2 } μ 2 + { κ c 2 + κ s 2 2 + ( γ e γ 0 κ s 2 ) 2 ( γ o 2 + γ e 2 ) κ c 2 κ s 4 } = 0 .
E x ( z ) = exp [ i k z ] m = E x ( m ) exp ( i m q z ) ,
E y ( z ) = exp [ i k z ] m = E y ( m ) exp ( i m q z ) ,
[ k e 2 ( k + m q ) 2 ] E x ( m ) + ω 2 c 2 n e 2 n o 2 2 p = , p 0 [ c p E x ( m p ) + s p E y ( m p ) ] = 0 ,
[ k o 2 ( k + m q ) 2 ] E y ( m ) + ω 2 c 2 n e 2 n o 2 2 p = , p 0 [ s p E x ( m p ) c p E y ( m p ) ] = 0 .
Δ ω PTN ( sin , full ) = 0.582 Δ ω Ch ( full ) .
κ L = Δ ω ( full ) 2 c n e + n o 2 L ,
Δ ω PTN ( rectangular , full ) = 0.637 Δ ω ( Ch , full ) .
ω 1 ω 2 = [ q c ( n e n o ) n 2 ] J 0 ( 2 θ s ) = Δ ω = ω 0 J 2 ( 2 θ s ) ( n e n o ) n ,

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