Abstract

We have worked out the diffraction pattern in the phase-grating mode of a cholesteric liquid crystal (cholesteric) with a pitch gradient. The pitch gradients considered are symmetric and asymmetric with respect to the sample center. For a uniform input beam, the intensity profile of each diffraction order becomes broad. Further, in the symmetric gradient, the profile of each order is irregular, while in the asymmetric gradient it is nearly flat. For a Gaussian input beam, for the symmetrically deformed structure the profile for each order is asymmetric, while for the asymmetrically deformed structure, the profiles just becomes broad. We find that even a 5% nonuniformity in the pitch can drastically alter the diffraction profiles.

© 2002 Optical Society of America

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References

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  1. P. J. Collings, J. S. Patel, Handbook of Liquid Crystal Research (Oxford U. Press, New York, 1997).
  2. S. Chandrasekar, Liquid Crystals (Cambridge U. Press, Cambridge, UK, 1992).
  3. P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).
  4. S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).
  5. R. Bartolino, N. Scaramuzza, “An optical effect of the mechanical deformation of the cholesteric ordering,” in Symmetries and Broken Symmetries in Condensed Matter Physics, N. Boccara, ed., Proceedings of the Colloque Pierre Curie Vol. 385 [Institut pour le Développment de la Science, l’Education et la Technologie (IDSET), Paris, 1981].
  6. D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
    [CrossRef]
  7. A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
    [CrossRef]
  8. E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
    [CrossRef] [PubMed]
  9. C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1936).
  10. K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
    [CrossRef]
  11. J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications (Prentice-Hall of India, New Delhi, 1995).
  12. O. D. Lavrentovich, S. V. Shiyanovskii, D. Voloschenko, “Fast beam steering cholesteric diffractive devices ,” in Optical Scanning: Design and Application, L. Beiser, S. F. Sagan, G. F. Marshall, eds., Proc. SPIE3787, 149–155 (1999).
    [CrossRef]
  13. S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, UK, 1995).
  14. K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
    [CrossRef] [PubMed]
  15. D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
    [CrossRef]

2000 (1)

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

1997 (1)

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

1995 (1)

D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
[CrossRef]

1994 (1)

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

1992 (1)

K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
[CrossRef]

1975 (1)

S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).

1968 (1)

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

1936 (1)

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1936).

Bartolino, R.

R. Bartolino, N. Scaramuzza, “An optical effect of the mechanical deformation of the cholesteric ordering,” in Symmetries and Broken Symmetries in Condensed Matter Physics, N. Boccara, ed., Proceedings of the Colloque Pierre Curie Vol. 385 [Institut pour le Développment de la Science, l’Education et la Technologie (IDSET), Paris, 1981].

Binet, C.

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

Bos, Ph.

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

Boucher, E.

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

Boudet, A.

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

Bourgerette, C.

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

Broer, D. J.

D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
[CrossRef]

Chandrasekar, S.

S. Chandrasekar, Liquid Crystals (Cambridge U. Press, Cambridge, UK, 1992).

Collings, P. J.

P. J. Collings, J. S. Patel, Handbook of Liquid Crystal Research (Oxford U. Press, New York, 1997).

De Gennes, P. G.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).

Dietz, R. E.

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

Kumar, P. B. S.

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
[CrossRef]

Lavrentovich, O. D.

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

O. D. Lavrentovich, S. V. Shiyanovskii, D. Voloschenko, “Fast beam steering cholesteric diffractive devices ,” in Optical Scanning: Design and Application, L. Beiser, S. F. Sagan, G. F. Marshall, eds., Proc. SPIE3787, 149–155 (1999).
[CrossRef]

Lipson, H.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, UK, 1995).

Lipson, S. G.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, UK, 1995).

Lub, J.

D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
[CrossRef]

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications (Prentice-Hall of India, New Delhi, 1995).

Mazkedian, S.

S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).

Meiboom, S.

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

Meixner, A. E.

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

Melone, S.

S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).

Mitov, M.

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

Mol, G. N.

D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
[CrossRef]

Nath, N. S. Nagendra

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1936).

Patel, J. S.

P. J. Collings, J. S. Patel, Handbook of Liquid Crystal Research (Oxford U. Press, New York, 1997).

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications (Prentice-Hall of India, New Delhi, 1995).

Prost, J.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).

Raman, C. V.

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1936).

Ranganath, G. S.

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
[CrossRef]

Rustichelli, F.

S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).

Sackmann, E.

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

Sah, Yuvaraj

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

Scaramuzza, N.

R. Bartolino, N. Scaramuzza, “An optical effect of the mechanical deformation of the cholesteric ordering,” in Symmetries and Broken Symmetries in Condensed Matter Physics, N. Boccara, ed., Proceedings of the Colloque Pierre Curie Vol. 385 [Institut pour le Développment de la Science, l’Education et la Technologie (IDSET), Paris, 1981].

Shiyanovskii, S. V.

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

O. D. Lavrentovich, S. V. Shiyanovskii, D. Voloschenko, “Fast beam steering cholesteric diffractive devices ,” in Optical Scanning: Design and Application, L. Beiser, S. F. Sagan, G. F. Marshall, eds., Proc. SPIE3787, 149–155 (1999).
[CrossRef]

Subacius, D.

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

Suresh, K. A.

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
[CrossRef]

Sydner, L. C.

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

Tannhauser, D. S.

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, UK, 1995).

Voloschenko, D.

O. D. Lavrentovich, S. V. Shiyanovskii, D. Voloschenko, “Fast beam steering cholesteric diffractive devices ,” in Optical Scanning: Design and Application, L. Beiser, S. F. Sagan, G. F. Marshall, eds., Proc. SPIE3787, 149–155 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

D. Subacius, S. V. Shiyanovskii, Ph. Bos, O. D. Lavrentovich, “Cholesteric gratings with field-controlled periods,” Appl. Phys. Lett. 71, 3323–3325 (1997).
[CrossRef]

Eur. Phys. J. E (1)

A. Boudet, C. Binet, M. Mitov, C. Bourgerette, E. Boucher, “Microstructure of variable pitch cholesteric films and its relationship with the optical properties,” Eur. Phys. J. E 2, 247–253 (2000).
[CrossRef]

J. Am. Chem. Soc. (1)

E. Sackmann, S. Meiboom, L. C. Sydner, A. E. Meixner, R. E. Dietz, “On the structure of the liquid crystalline state of cholesterol derivatives,” J. Am. Chem. Soc. 90, 3567–3569 (1968).
[CrossRef] [PubMed]

J. Phys. (Paris) Suppl. (1)

S. Mazkedian, S. Melone, F. Rustichelli, “On light diffraction by cholesteric crystals with a pitch gradient,” J. Phys. (Paris) Suppl. C1, 283–287 (1975).

Liq. Cryst. (1)

K. A. Suresh, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in twisted liquid-crystalline media—phase grating mode,” Liq. Cryst. 11, 73–82 (1992).
[CrossRef]

Nature (1)

D. J. Broer, J. Lub, G. N. Mol, “Wide band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378, 467–469 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

K. A. Suresh, Yuvaraj Sah, P. B. S. Kumar, G. S. Ranganath, “Optical diffraction in chiral smectic-C liquid crystals,” Phys. Rev. Lett. 72, 2863–2866 (1994).
[CrossRef] [PubMed]

Proc. Indian Acad. Sci. Sect. A (1)

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–412 (1936).

Other (7)

J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications (Prentice-Hall of India, New Delhi, 1995).

O. D. Lavrentovich, S. V. Shiyanovskii, D. Voloschenko, “Fast beam steering cholesteric diffractive devices ,” in Optical Scanning: Design and Application, L. Beiser, S. F. Sagan, G. F. Marshall, eds., Proc. SPIE3787, 149–155 (1999).
[CrossRef]

S. G. Lipson, H. Lipson, D. S. Tannhauser, Optical Physics (Cambridge U. Press, Cambridge, UK, 1995).

R. Bartolino, N. Scaramuzza, “An optical effect of the mechanical deformation of the cholesteric ordering,” in Symmetries and Broken Symmetries in Condensed Matter Physics, N. Boccara, ed., Proceedings of the Colloque Pierre Curie Vol. 385 [Institut pour le Développment de la Science, l’Education et la Technologie (IDSET), Paris, 1981].

P. J. Collings, J. S. Patel, Handbook of Liquid Crystal Research (Oxford U. Press, New York, 1997).

S. Chandrasekar, Liquid Crystals (Cambridge U. Press, Cambridge, UK, 1992).

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of a cholesteric with its twist axis along the z direction, showing the projection of the director as viewed normal to the twist axis. The twist axis is aligned parallel to the glass plates. A plane wave front, incident in a direction Ki normal to the twist axis of the sample with its electric vector E perpendicular to the twist axis, encounters a varying refractive index. In this phase-grating mode, the emergent wave front is corrugated and results in diffraction. Here t is the sample thickness and L is the lateral width of the sample.

Fig. 2
Fig. 2

Exaggerated view of the refractive index μl as a function of position z. (a) In a type I cholesteric the refractive index profile is symmetric with respect to the center of the sample. (b) In a type II cholesteric the refractive index profile is asymmetric with respect to the center of the sample. In both cases the center of the sample is indicated by the arrow. The values of μe and μo used in our computations are 1.605 and 1.535, respectively.

Fig. 3
Fig. 3

Diffraction pattern for an input beam of uniform intensity. (a) A uniform cholesteric of pitch 20 μm. Here the angle of diffraction for the first order in the cholesteric medium is 1.43°. (b) A type II cholesteric, where the pitch varies from 20 μm at one end of the sample to 22 μm at the other end. (c) Type Ia cholesteric, where the pitch varies from 21 μm at the sample center to 20 μm at the sample ends. (d) Type Ib cholesteric, where the pitch varies from 20 μm at the sample center to 21 μm at the sample ends. The values of the parameters used in the calculations are as follows: birefringence, Δμ=0.07; lateral sample width, L=100 pitches; thickness, t=15 μm; wavelength of light, λ=0.633 μm.

Fig. 4
Fig. 4

Diffraction pattern for a Gaussian intensity profile of half-width σ/2. The results shown correspond, one to one, to the same cholesteric structures considered in Fig. 3. In (c) and (d) the inset shows a magnified view of the fifth diffraction order highlighting the asymmetry in the intensity profiles. Here the parameter σ used in the calculations is equal to 1 mm.

Fig. 5
Fig. 5

Diffraction patterns for samples with 20% pitch gradient. (a), (b), and (c) correspond to type II, type Ia, and type Ib cholesterics, respectively, illuminated by uniform input beams. (d), (e), and (f) correspond to the same structures when they are illuminated with Gaussian input beams. Here the base pitch of the cholesteric is again 20 μm, and the values of the other parameters are the same as given in Fig. (3).

Equations (17)

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

1μu2=sin2(α)μe2+cos2(α)μo2
nx=cos(q0z+ϕ),
ny=sin(q0z+ϕ),
nz=0,
nx=cos(qz+ϕ),
ny=sin(qz+ϕ),
nz=0,
q=(q0+z),
1μl2=ny2μe2+nx2μo2.
U(z)=exp(2πiμlt/λ),
AU(k)=-L/2+L/2exp(2πiμlt/λ)exp(-ikz)dz,
AG(k)=-L/2+L/2exp[-(z/σ)2]exp(2πiμlt/λ)exp(-ikz)dz,
Fd=K222θz-q02,
θ=q0z+u(z);
u(z)=q=qminqmaxuqexp(iqz),
qmin=2πL,qmax=2πd,
|uq|2=KBTq2K22.

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