See the 2 × 2 Jones-matrix analogs of these equations in Ref. 1 [Eqs. (2. 6), (2. 7), (2. 17), and (2. 13)].

The inverse of M_{z} is derived from M_{z} by changing the sign of z and the differentiation is elementary.

di._{+} represents the isotropic refractive properties of the medium.

1t can be easily verified that this matrix is the 4 × 4 equivalent of the 2 × 2 Jones matrix N obtained by adding the eight differential Jones matrices listed in Table I, in accordance with the transformation specified by Eqs. (26) and (29). This provides a check on the latter transformation.

FromEqs. (45) and (47) itcanbe concluded thatforahomogeneous medium (1) the eigenvectors of the intensive differential matrix m and the extensive Mueller matrix M are the same, and (2) an eigenvalue *Ʋ*_{M} of M is related to the corresponding eigenvalue *Ʋ*_{m} of m by *Ʋ*_{M}=exp(*Ʋ*_{m}z). In Ref. 1, Jones provides a different proof of the similar relationships between the eigenvectors and eigenvalues of his intensive and extensive 2×2 matrices.

See, for example, C. R. Wylie, Jr., Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 487.

This is probably the first time that the Mueller-matrix eigenvalue problem has been considered and the significance of its eigenvectors and eigenvalues examined.

With direction cosines (*β*_{n}, γ_{n}, δ_{n})

Because the degree of polarization cannot exceed one, c is limited to the range -1/*ω*< *c* < 1/*ω*. Values of c outside this range yield nonphysical, but mathematically acceptable, eigenvectors.

Notice that because *S*_{0}=1, S_{i}=s_{i} (*i* =1, 2, 3).

See, for example, Ref. 2 and the references cited therein.

The results of previous treatments can be used to account for the propagation of partially polarized light only by decomposing such light into its totally polarized and unpolarized components. The present development is direct in its handling of partial polarization and, more importantly, is capable of analyzing depolarization effects as may be caused, for example, by small random perturbations in the molecular ordering of the liquid crystal.

See Ref. 11, Chap. 2.

It is interesting to note that, for any initial polarization, the Stokes parameters *S*_{1} and *S*_{2} have three spatial-frequency components: one at double the spatial frequency of the helical structure (2*ρ*) and the other two [(2*ρ*+*ζ*) and (2*ρ*- *ζ*)] are upand down-shifted from this (center) frequency by the same amount (*ζ*). The Stokes parameter *S*_{3} has a constant (dc) component and one spatial-frequency component (*ζ*). If we derive the ellipticity and azimuth of the polarization ellipse of the wave from the Stokes parameters, we will find that they have the same behavior as a function of distance as has been pointed out in Ref. 2.

See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsonietry and Polarized Light (North-Holland, Amsterdam, 1977).

For simplicity, we drop the argument or subscript z, which indicates z dependence, from S and m. Equations (3) and (4) are the 4×4-matrix analogs of Eqs. (2. 10) and (2. 2), respectively, in Ref. 1.