Abstract

Trajectories are given that describe the evolution of the ellipse of polarization in the complex plane for light propagating in a homogeneous anisotropic medium and along the helical axis of a cholesteric liquid crystal. For the general homogeneous anisotropic medium that exhibits combined birefringence and dichroism the trajectory is a spiral that converges to the low-absorption eigenpolarization. For pure birefringence the trajectory becomes a complete circle that encloses one eigenpolarization, whereas for pure dichroism the trajectory becomes an arc of a circle that ends at the low-absorption eigenstate. The case of a cholesteric (or twisted nematic) liquid crystal leads to an interesting family of trajectories that can be considered as distorted hypo- or epicycloids. These trajectories are nonrepetitive (open) and may show multilobes or branches depending upon the initial polarization and the properties of the liquid crystal. Graphs are also presented where the ellipticity and azimuth are plotted separately as functions of distance along the helical axis.

© 1973 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 1252 (1972).
    [CrossRef]
  2. R. C. Jones, J. Opt. Soc. Am. 38, 671 (1948).
    [CrossRef]
  3. The cholesteric liquid crystal problem has also been treated by C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Crystallogr. 4, 219 (1951); G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968); D. W. Berreman, T. J. Scheffer, Mol. Crystallogr. Liquid Crystallogr. 11, 395 (1970), J. Opt. Soc. Am. 62, 502 (1972); A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971), Optics Commun. 3, 369 (1971). This is not an exhaustive list.
    [CrossRef]
  4. The proof follows from finding the ratios A/C and B/D using Eq. (8) and noticing that the condition of their independence of χ0 leads to the definition of β in Eq. (3). That these ratios give the eigenpolarizations can be checked by substitution in Eq. (7).
  5. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [CrossRef]
  6. R. M. A. Azzam, N. M. Bashara, Optics Commun. 5, 319 (1972).
    [CrossRef]
  7. A conformal transformation preserves the shape of any curve in the immediate vicinity of a point in the complex plane. The spiraling behavior of exp(i2βz) around the origin (or the point at infinity) is therefore reproduced by χ(z,χ0) around the low-absorption eigenpolarization.
  8. P. K. Rees, Analytic Geometry (Prentice-Hall, Englewood Cliffs, N. J., 1970), p. 177.

1972 (3)

1948 (1)

1933 (1)

The cholesteric liquid crystal problem has also been treated by C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Crystallogr. 4, 219 (1951); G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968); D. W. Berreman, T. J. Scheffer, Mol. Crystallogr. Liquid Crystallogr. 11, 395 (1970), J. Opt. Soc. Am. 62, 502 (1972); A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971), Optics Commun. 3, 369 (1971). This is not an exhaustive list.
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

Jones, R. C.

Oseen, C. W.

The cholesteric liquid crystal problem has also been treated by C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Crystallogr. 4, 219 (1951); G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968); D. W. Berreman, T. J. Scheffer, Mol. Crystallogr. Liquid Crystallogr. 11, 395 (1970), J. Opt. Soc. Am. 62, 502 (1972); A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971), Optics Commun. 3, 369 (1971). This is not an exhaustive list.
[CrossRef]

Rees, P. K.

P. K. Rees, Analytic Geometry (Prentice-Hall, Englewood Cliffs, N. J., 1970), p. 177.

J. Opt. Soc. Am. (3)

Optics Commun. (1)

R. M. A. Azzam, N. M. Bashara, Optics Commun. 5, 319 (1972).
[CrossRef]

Trans. Faraday Soc. (1)

The cholesteric liquid crystal problem has also been treated by C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933); Hl. De Vries, Acta Crystallogr. 4, 219 (1951); G. H. Conners, J. Opt. Soc. Am. 58, 875 (1968); D. W. Berreman, T. J. Scheffer, Mol. Crystallogr. Liquid Crystallogr. 11, 395 (1970), J. Opt. Soc. Am. 62, 502 (1972); A. S. Marathay, J. Opt. Soc. Am. 61, 1363 (1971), Optics Commun. 3, 369 (1971). This is not an exhaustive list.
[CrossRef]

Other (3)

The proof follows from finding the ratios A/C and B/D using Eq. (8) and noticing that the condition of their independence of χ0 leads to the definition of β in Eq. (3). That these ratios give the eigenpolarizations can be checked by substitution in Eq. (7).

A conformal transformation preserves the shape of any curve in the immediate vicinity of a point in the complex plane. The spiraling behavior of exp(i2βz) around the origin (or the point at infinity) is therefore reproduced by χ(z,χ0) around the low-absorption eigenpolarization.

P. K. Rees, Analytic Geometry (Prentice-Hall, Englewood Cliffs, N. J., 1970), p. 177.

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Figures (12)

Fig. 1
Fig. 1

(a) The function exp(i2βz) for β real and positive gives the unit circle in the complex planes. The rotational sense is reversed when β is negative. (b) The trajectory of χ(z,χ0) is an image through a bilinear transformation [Eq. (7)] of the unit circle of (a). χ(z,χ0) is a circle that is traversed periodically as z is increased starting from the initial state χ0 at z = 0.

Fig. 2
Fig. 2

(a) The function exp(i2βz) when β is purely imaginary and positive (left) or negative (right). (b) χ(z,χ0), which is the image of exp(i2βz) in (a), follows an arc of a circle that starts at initial state χ0(z = 0) and converges asymptotically to the final state χf(z = ∞). χf is the low-absorption eigenpolarization.

Fig. 3
Fig. 3

(a) The function exp(i2βz) for β complex (β = βr + i) is a logarithmic spiral. Reversing the sign of βi causes the spiral to expand out to infinity. The sense of the spiral is clockwise (instead of counterclockwise) when βr is negative. (b) χ(z,χ0), being an image of the spiral of (a), is also a spiral. The final state χf is the low-absorption eigenpolarization.

Fig. 4
Fig. 4

(a) The computed evolution of the ellipse of polarization [χ(z,χ0)] in an elliptically birefringent medium with eigenpolarizations χe1 and χe2 for eight different initial polarization states A, B.,… and H. The N matrix of the medium has the form of Neb in Eq. (14) with A = 0.75 + i.25 and B = 0.875 + i.375. (b) The variation of the ellipticity e (△) and azimuth a (○) with distance z along the direction of propagation in the elliptically birefringent medium corresponding to the trajectory that starts at H in (a). Note that both functions are periodic with z.

Fig. 5
Fig. 5

(a) Same as in Fig. 4(a) for an elliptically dichroic medium whose N matrix has the form of Ned in Eq. (14) with p = 2, q = 1, and R = 0.866 + i. 5. A, B, C, D indicate four different initial states and χf, the final state, corresponds to the low-absorption eigenpolarization. (b) Same as in Fig. 4(b) for the trajectory that starts at C in (a). Note the aperiodic nature of the curves and their asymptotic the approach to the ellipticity and azimuth of the low-absorption eigenpolarization.

Fig. 6
Fig. 6

(a) Same as in Figs. 4(a) and 5(a) for the general homogeneous anisotropic medium that exhibits combined birefringence and dichroism. The N matrix was chosen arbitrarily with elements n11 = 1 + i2, n12 = 2 + i3, n21 = 3 + i4, and n22 = 4 + i5. Note the spiraling convergence to the low-absorption eigenpolarization χf for the four different initial polarizations A, B, C, and D. (b) Same as in Figs. 4(b) trajectory that starts at B in (a). Note the damped oscillatory convergence to the final ellipticity and azimuth.

Fig. 7
Fig. 7

The trajectories of u(z,u0) describing the evolution of the polarization ellipse of light propagating along the helical axis of a cholesteric liquid crystal for different initial states, referenced to a space-rotating coordinate system that coincides with the principal axes of the molecular planes K1K2 and are the two privileged polarization ellipses whose axes are forced to remain in alignment with the principal axes of the molecular planes with their ellipticities unchanged. The complex-plane representation in this figure as well as in (a) of Figs. 812 uses the left- and right circular polarizations as basis states and are represented by the origin and point of infinity, respectively.

Fig. 8
Fig. 8

(a) The evolution of the polarization ellipse along the helical axis of a cholesteric liquid crystal with α = g0 = 1 in a space-fixed coordinate system for an initial state χ0 = −i. Two circles can be drawn with center at the origin between which the trajectory of χ(z,χ0) is confined. The nonperiodic nature of the trajectory can be easily seen. (b) The evolution of the ellipticity e (△) and azimuth a (○) along the helical axis for the trajectory of (a). The ellipticity is periodic and the azimuth is not. See text for an explanation of the apparent discontinuities in azimuth both in this figure and in the ones to come.

Fig. 9
Fig. 9

(a) Same as in Fig. 8(a) for a different initial state χ0 = −1. The trajectory of χ(z,χ0) has overlapping multilobes (leaves). Note that the limiting inner circle has collapsed to a point (the origin). (b) The ellipticity e (△) and azimuth a (○) for the trajectory of (a).

Fig. 10
Fig. 10

(a) Same as in Figs. 8(a) and 9(a) for still another initial state, χ0 = 1. In this case u(z,u0) becomes the straight line parallel to the imaginary axis bisecting the distance between K1 and K2 in Fig. 7 [see Eq. (17)]. The trajectory of χ(z,χ0) has multibranches with the limiting outer circle at infinity. Points marked with the same number join smoothly at infinity. (b) The ellipticity e (△) and azimuth a (○) for the trajectory of (a).

Fig. 11
Fig. 11

(a) Same as in Figs. 8(a), 9(a), and 10(a) for a liquid crystal with the same pitch (α = 1) but with different molecular birefringence (χ0 = 0.5). The initial polarization state is χ0 = 1. This trajectory clearly resembles a hypocycloid, perhaps better than any other figure in this group of trajectories. (b) The ellipticity e (△) and azimuth a (○) for the liquid crystal and trajectory of (a).

Fig. 12
Fig. 12

(a) Same as in Figs. 8(a), 9(a), and 10(a) except that the crystal now has a different pitch (α = 0.5) with g0 the same (g0 = 1). The initial state is χ0 = 0. (b) The ellipticitty e (△) and azimuth a (○) for the liquid crystal and trajectory of (a).

Equations (19)

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d χ / d z = n 12 χ 2 + ( n 22 - n 11 ) χ + n 21 ,
χ ( z , χ 0 ) = { [ β - 1 2 ( n 11 - n 22 ) tan β z ] χ 0 + ( n 21 tan β z ) } / { ( n 12 tan β z ) χ 0 + [ β + 1 2 ( n 11 - n 22 ) tan β z ] } ,
β = [ - 1 4 ( n 11 - n 22 ) 2 - n 12 n 21 ] 1 / 2
χ ¯ ( z , χ 0 ) = [ β - i α tan β z ) χ 0 + ( - i g 0 tan β z ) ] [ ( - i g 0 tan β z ) χ 0 + ( β + i α tan β z ) ] exp ( i 2 α z ) ,
α = 2 π / p ,
β = g 0 / [ ( α / g 0 ) 2 + 1 ] 1 / 2 .
tan β z = i [ 1 - exp ( i 2 β z ) ] / [ 1 + exp ( i 2 β z ) ] ,
χ ( z , χ 0 ) = [ A + B exp ( i 2 β z ) ] / [ C + D exp ( i 2 β z ) ] ,
A = i n 21 + [ β - 1 2 i ( n 11 - n 22 ) ] χ 0 , B = - i n 21 + [ β + 1 2 i ( n 11 - n 22 ) ] χ 0 , C = [ β + 1 2 i ( n 11 - n 22 ) ] + i n 12 χ 0 , D = [ β - 1 2 i ( n 11 - n 22 ) ] - i n 12 χ 0 .
χ e 1 = A / C , χ e 2 = B / D .
χ e 1 , 2 = ( 1 / 2 n 12 ) { ( n 22 - n 11 ) ± [ ( n 22 - n 11 ) 2 + 4 n 12 n 21 ] 1 / 2 } .
d = π / β .
χ f = A / C = χ e 1 ,
χ f = B / D = χ e 2 .
Im [ 1 4 ( Tr N ) 2 - ( Det N ) ] = 0 ,
N eb = [ A B - B * A * ] , N ed = [ p R R * q ] ,
u ( z , u 0 ) = [ A + B exp ( i 2 β z ) ] / [ C + D exp ( i 2 β z ) ] ,
A = g 0 + ( α + β ) χ 0 , B = - g 0 - ( α - β ) χ 0 , C = - ( α - β ) + g 0 χ 0 , D = ( α + β ) - g 0 χ 0 .
Re χ 0 = Re u 0 = 1 2 ( k 1 + k 2 ) = α / g 0

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