Abstract

The reflection spectrum of light from the surface of a cholesteric liquid crystal film has been theoretically studied by including the absorption of light by the material. The intensity of reflection is found as a function of the wavelength and the angle of incidence as well as the angle of reflection near the circular dichroism region. A good agreement has been obtained with the available experimental data.

© 1971 Optical Society of America

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References

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  1. G. H. Brown, W. G. Shaw, Chem. Rev. 57, 1049 (1957).
    [Crossref]
  2. J. P. Mathieu, Handbuch der Physik 28, 421 (1957).
  3. J. L. Fergason, Appl. Opt. 7, 1729 (1968).
    [Crossref] [PubMed]
  4. A. Saupe, Angew. Chem. Int. Ed. Eng. 7, 97 (1968).
    [Crossref]
  5. G. Durand, Compt. Rend. B264, 1251 (1967).
  6. J. L. Fergason, Mol. Cryst. 1, 293 (1966).
    [Crossref]
  7. J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
    [Crossref]
  8. C. G. Darwin, Phil. Mag. 27, 675 (1914).
  9. R. W. James, Optical Principles of Diffraction of X-Rays (G. Bell and Sons, London, 1948).
  10. S. Chandrasekhar, K. N. Srinivasa Rao, Acta Cryst. A24, 445 (1968).
  11. W. Haas, J. E. Adams, Appl. Opt. 7, 1203 (1968).
    [Crossref] [PubMed]
  12. J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
    [Crossref]

1969 (2)

J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
[Crossref]

J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
[Crossref]

1968 (4)

S. Chandrasekhar, K. N. Srinivasa Rao, Acta Cryst. A24, 445 (1968).

W. Haas, J. E. Adams, Appl. Opt. 7, 1203 (1968).
[Crossref] [PubMed]

J. L. Fergason, Appl. Opt. 7, 1729 (1968).
[Crossref] [PubMed]

A. Saupe, Angew. Chem. Int. Ed. Eng. 7, 97 (1968).
[Crossref]

1967 (1)

G. Durand, Compt. Rend. B264, 1251 (1967).

1966 (1)

J. L. Fergason, Mol. Cryst. 1, 293 (1966).
[Crossref]

1957 (2)

G. H. Brown, W. G. Shaw, Chem. Rev. 57, 1049 (1957).
[Crossref]

J. P. Mathieu, Handbuch der Physik 28, 421 (1957).

1914 (1)

C. G. Darwin, Phil. Mag. 27, 675 (1914).

Adams, J. E.

J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
[Crossref]

J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
[Crossref]

W. Haas, J. E. Adams, Appl. Opt. 7, 1203 (1968).
[Crossref] [PubMed]

Brown, G. H.

G. H. Brown, W. G. Shaw, Chem. Rev. 57, 1049 (1957).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, K. N. Srinivasa Rao, Acta Cryst. A24, 445 (1968).

Darwin, C. G.

C. G. Darwin, Phil. Mag. 27, 675 (1914).

Durand, G.

G. Durand, Compt. Rend. B264, 1251 (1967).

Fergason, J. L.

Haas, W.

J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
[Crossref]

J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
[Crossref]

W. Haas, J. E. Adams, Appl. Opt. 7, 1203 (1968).
[Crossref] [PubMed]

James, R. W.

R. W. James, Optical Principles of Diffraction of X-Rays (G. Bell and Sons, London, 1948).

Mathieu, J. P.

J. P. Mathieu, Handbuch der Physik 28, 421 (1957).

Saupe, A.

A. Saupe, Angew. Chem. Int. Ed. Eng. 7, 97 (1968).
[Crossref]

Shaw, W. G.

G. H. Brown, W. G. Shaw, Chem. Rev. 57, 1049 (1957).
[Crossref]

Srinivasa Rao, K. N.

S. Chandrasekhar, K. N. Srinivasa Rao, Acta Cryst. A24, 445 (1968).

Wysocki, J.

J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
[Crossref]

J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
[Crossref]

Acta Cryst. (1)

S. Chandrasekhar, K. N. Srinivasa Rao, Acta Cryst. A24, 445 (1968).

Angew. Chem. Int. Ed. Eng. (1)

A. Saupe, Angew. Chem. Int. Ed. Eng. 7, 97 (1968).
[Crossref]

Appl. Opt. (2)

Chem. Rev. (1)

G. H. Brown, W. G. Shaw, Chem. Rev. 57, 1049 (1957).
[Crossref]

Compt. Rend. (1)

G. Durand, Compt. Rend. B264, 1251 (1967).

Handbuch der Physik (1)

J. P. Mathieu, Handbuch der Physik 28, 421 (1957).

J. Chem. Phys. (1)

J. E. Adams, W. Haas, J. Wysocki, J. Chem. Phys. 50, 2458 (1969).
[Crossref]

Mol. Cryst. (1)

J. L. Fergason, Mol. Cryst. 1, 293 (1966).
[Crossref]

Phil. Mag. (1)

C. G. Darwin, Phil. Mag. 27, 675 (1914).

Phys. Rev. Lett. (1)

J. E. Adams, W. Haas, J. Wysocki, Phys. Rev. Lett. 22, 92 (1969).
[Crossref]

Other (1)

R. W. James, Optical Principles of Diffraction of X-Rays (G. Bell and Sons, London, 1948).

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Figures (4)

Fig. 1
Fig. 1

Geometrical representation of reflection in a disturbed/undisturbed film (from Adams et al.7).

Fig. 2
Fig. 2

The reflection of light as a function of wavelength for a typical case in which the reflection width is small. Dotted curve is the result of Chandrasekhar’s neglecting the absorption. Q = 0.1762, H/Q = 0.0204. Experimental points are taken from Fergason.6

Fig. 3
Fig. 3

The reflection as a function of wavelength for a typical case of large reflection width. Dotted curve is the result of Chandrasekhar’s neglecting the absorption. Q = 0.22, H/Q = 0.294. The experimental points are taken from Adams et al.7

Fig. 4
Fig. 4

The reflection as a function of the scattered direction. Dotted curve is the result of Chandrasekhar’s neglecting the absorption. Q = 0.076, H/Q = 0.2. The experimental points are taken from Adams et al.7

Equations (28)

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S r = - i Q T r + ( 1 - H - i Q 0 ) exp ( - i ϕ ) S r + 1 ,
T r + 1 exp ( i ϕ ) = ( 1 - H - i Q 0 ) T r - i Q exp ( - i ϕ ) S r + 1 .
S 0 / T 0 = - i Q / [ 1 - x exp ( - i ϕ ) ( 1 - H - i Q 0 ) ] ,
( 1 - H - i Q 0 ) exp ( - i ϕ ) ( x + 1 / x ) = 1 + Q 2 exp ( - 2 i ϕ ) + ( 1 - H - i Q 0 ) 2 exp ( - 2 i ϕ ) .
2 μ P sin θ B = n λ 0 , n = 1 , 2 , .
2 π μ P sin θ / λ = n π + ,
= 2 π [ 1 - ( λ 0 sin θ / λ sin θ B ) ] .
For different λ , θ = θ B , = 2 π ( λ - λ 0 ) / λ .
= 2 π [ 1 - ( λ 0 sin θ / λ sin θ B ) ] .
sin θ B = cos 1 2 [ sin - 1 ( sin ϕ i / μ ) + sin - 1 ( sin ϕ s / μ ) ] ,
sin θ B = sin 1 2 [ sin - 1 ( sin ϕ i / μ ) - sin - 1 ( sin ϕ s / μ ) ] .
x = ( - 1 ) n ( 1 - δ ) = ( - 1 ) n [ 1 - ( l + i m ) ]
ϕ = n π + .
S 0 / T 0 = - i Q / { 1 - [ 1 - ( l + i m ) ] × exp ( - i ) ( 1 - H - i Q 0 ) }
| S 0 T 0 | 2 = Q 2 { 1 - ( 1 - H ) [ ( 1 - l ) cos - m sin ] } 2 + ( 1 - H ) 2 [ ( 1 - l ) sin + m cos ] 2 .
δ 2 + δ ( p - 2 ) - ( p - 2 ) = 0 ,
p = cos ( 2 + H 2 + Q 2 ) + i sin ( 2 H - Q 2 + H 2 ) ,
δ = 1 2 [ ( 2 - p ) ± ( p 2 - 4 ) 1 2 ] = 1 2 [ 2 - ( a + i b ) ± ( c + i d ) ] = 1 2 [ ( 2 - a ± c ) - i b ( 1 a / c ) ] ,
c = { [ ( a 2 - b 2 - 4 ) 2 + 4 a 2 b 2 ] 1 2 + ( a 2 - b 2 - 4 ) } 1 2 / 2 1 2 .
c a b / a 2 - b 2 - 4 1 2 ,
δ = 1 2 ( 2 - a ± c - i b ± i a 2 - b 2 - 4 1 2 ) .
| S 0 T 0 | 2 = 1 / [ 1 + 2 x - y 2 + ( 2 x - y ) 2 8 - x y 2 ] 2 ,
| S 0 T 0 | 2 = 1 / [ 1 + 2 1 2 ( 2 x - y ) 1 2 + ( 2 x - y ) + ( 2 x - y ) 3 2 / 2 1 2 + 5 y 2 / 48 ( 2 ) 1 2 ( 2 x - y ) 1 2 + ( 1 / 96 ) ( 96 x 2 - 96 x y - y 2 ) ] .
| S 0 T 0 | 2 = 1 / [ ( H Q + l Q ) 2 + 2 Q 2 ( 1 + H Q · l Q ) 2 ] ,
l Q = 1 2 1 2 { [ ( 1 + H 2 Q 2 - 2 Q 2 ) 2 + 4 2 H 2 Q 4 ] 1 2 + ( 1 + H 2 Q 2 - 2 Q 2 ) } 1 2 .
l Q = ( 1 + H 2 Q 2 ) 1 2 , | S 0 T 0 | 2 = 1 / ( H Q + l Q ) 2 .
l Q = 1 2 1 2 [ ( H 4 Q 4 + 4 H 2 Q 2 ) 1 2 + H 2 Q 2 ] 1 2 ,
| S 0 T 0 | 2 = 1 / [ ( H Q + l Q ) 2 + ( 1 + H Q · Q l ) 2 ] .

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