Abstract

We extend the scope of the Mueller calculus to parallel that established by Jones for his calculus. We find that the Stokes vector S of a light beam that propagates through a linear depolarizing anisotropic medium obeys the first-order linear differential equation dS/dz = mS, where z is the distance traveled along the direction of propagation and m is a 4 × 4 real matrix that summarizes the optical properties of the medium which influence the Stokes vector. We determine the differential matrix m for eight basic types of optical behavior, find its form for the most general anisotropic nondepolarizing medium, and determine its relationship to the complex 2 × 2 differential Jones matrix. We solve the Stokes-vector differential equation for light propagation in homogeneous nondepolarizating media with arbitrary absorptive and refractive anisotropy. In the process, we solve the differential-matrix and Mueller-matrix eigenvalue equations. To illustrate the case of inhomogeneous anisotropic media, we consider the propagation of partially polarized light along the helical axis of a cholesteric or twisted-nematic liquid crystal. As an example of depolarizing media, we consider light propagation through a medium that tends to equalize the preference of the state of polarization to the right and left circular states.

© 1978 Optical Society of America

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References

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  1. R. C. Jones, “A New Calculus for the Treatment of Optical Systems. VII, Properties of the N-Matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  2. An alternative related description is that based on the complex polarization variable χ(equal to the ratio of the complex amplitudes of two orthogonal components of E) which obeys Riccati’s differential equation. See R. M. A. Azzam and N. M. Bashara, “Simplified Approach to the Propagation of Polarized Light in Anisotropic Media—Application to Liquid Crystals,” J. Opt. Soc. Am. 62, 1252–1257 (1972).
    [CrossRef]
  3. See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  4. 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.
  5. R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, “Trajectories Describing the Evolution of Polarized Light in Homogeneous Anisotropic Media and Liquid Crystals,” Appl. Opt. 12, 764–771 (1973); B. E. Merrill, R. M. A. Azzam, and N. M. Bashara, “Numerical Solution for the Evolution of the Ellipse of Polarization in Inhomogeneous Anisotropic Media,” J. Opt. Soc. Am. 64, 731–733 (1974).
    [CrossRef] [PubMed]
  6. (See the 2 × 2 Jones-matrix analogs of these equations in Ref. 1 [Eqs. (2. 6), (2. 7), (2. 17), and (2. 13)].
  7. The inverse of Mz is derived from Mz by changing the sign of z and the differentiation is elementary.
  8. d+irepresents the isotropic refractive properties of the medium.
  9. It 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.
  10. From Eqs. (45) and (47) it can be concluded that for a homogeneous medium (1) the eigenvectors of the intensive differential matrix m and the extensive Mueller matrix M are the same, and (2) an eigenvalue vM of M is related to the corresponding eigenvalue vm of m by vM= exp(vmz). 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.
  11. See, for example, C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 487.
  12. This is probably the first time that the Mueller-matrix eigenvalue problem has been considered and the significance of its eigenvectors and eigenvalues examined.
  13. With direction cosines (βn, γn, δn).
  14. 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.
  15. Notice that because S0 = 1, Si=si(i= 1, 2, 3).
  16. See, for example, Ref. 2 and the references cited therein.
  17. 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.
  18. See Ref. 11, Chap. 2.
  19. It is interesting to note that, for any initial polarization, the Stokes parameters S1 and S2 have three spatial-frequency components: one at double the spatial frequency of the helical structure (2ρ) and the other two [(2ρ+ ζ) and (2ρ− ζ)] are up- and down-shifted from this (center) frequency by the same amount (ζ). The Stokes parameter S3 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.
  20. D. W. Berreman, “Optics in Stratified Anisotropic Media: 4 × 4-Matrix Formulation,” J. Opt. Soc. Am. 62, 502–510 (1972) and references therein.
    [CrossRef]
  21. R. C. Jones, “New Calculus for the Treatment of Optical Systems. VIII. Electromagnetic Theory,” J. Opt. Soc. Am. 46, 126–131 (1956).
    [CrossRef]

1973 (1)

1972 (2)

1956 (1)

1948 (1)

Azzam, R. M. A.

Bashara, N. M.

Berreman, D. W.

Jones, R. C.

Merrill, B. E.

Wylie, C. R.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Other (16)

See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry 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.

(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 Mz is derived from Mz by changing the sign of z and the differentiation is elementary.

d+irepresents the isotropic refractive properties of the medium.

It 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.

From Eqs. (45) and (47) it can be concluded that for a homogeneous medium (1) the eigenvectors of the intensive differential matrix m and the extensive Mueller matrix M are the same, and (2) an eigenvalue vM of M is related to the corresponding eigenvalue vm of m by vM= exp(vmz). 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, 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 (βn, γn, δn).

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 S0 = 1, Si=si(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 S1 and S2 have three spatial-frequency components: one at double the spatial frequency of the helical structure (2ρ) and the other two [(2ρ+ ζ) and (2ρ− ζ)] are up- and down-shifted from this (center) frequency by the same amount (ζ). The Stokes parameter S3 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.

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Figures (5)

FIG. 1
FIG. 1

Partially polarized light is assumed to propagate along the z axis of an xyz Cartesian coordinate system in an anisotropic medium that may exhibit depolarization. S(z) and S(z + Δz) are the Stokes vectors of the beam at two transverse planes located at z and z + Δz, respectively.

FIG. 2
FIG. 2

The normalized Stokes parameters s1, s2, s3 [Eqs. (6a)] constitute the components of a vector s that represents the state of polarization of partially polarized light completely. |s| gives the degree of polarization of the beam while the angular spherical coordinates ( 1 2 π - 2 ) and 2θ determine the ellipticity angle and azimuth θ of the polarization ellipse of the totally polarized component of the beam. The trajectory C of the endpoint of s gives an interesting graphical representation of the evolution of the state of polarization as light propagates through an anisotropic medium. The unit sphere in the s1s2s3 space is the Poincaré sphere (PS).

FIG. 3
FIG. 3

(a) The propagation of initially unpolarized light through a nondepolarizing medium with arbitrary absorptive anisotropy is represented by the radial (linear) trajectory C. I and F represent the initial and final unpolarized and totally polarized states, respectively. F corresponds to the low-absorption eigenpolarization of the medium and is asymptotically approached as the distance of propagation z is increased. (b) The degree of polarization |s| of initially unpolarized light grows as a hyperbolic-tangent function of the distance z traveled by such light in a nondepolarizing medium with arbitrary absorptive anisotropy.

FIG. 4
FIG. 4

The propagation of initially right circularly polarized light (I) through a nondepolarizing medium with arbitrary refractive anisotropy is represented by the circle C on the Poincaré sphere (PS) The center point E of the spherical cap bounded by C represents one of the two orthogonal eigenpolarizations of the medium.

FIG. 5
FIG. 5

The propagation of light through a depolarizing medium with the differential matrix given by Eq. (81) is represented by a vertical straight line (parallel to the s3 axis) through the point I which represents an arbitrary initial state. The point F which represents the final state of partial linear polarization is in the equatorial (s1s2) plane.

Tables (1)

Tables Icon

TABLE I The differential 4 × 4 matrix m and Jones 2 × 2 matrix N for eight basic types of optical behavior.a

Equations (94)

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S ( z + Δ z ) = M z , Δ z S ( z ) ,
S ( z + Δ z ) - S ( z ) = ( M z , Δ z - I ) S ( z ) ,
d S / d z = mS ,
m = lim Δ z 0 M z , Δ z - I Δ z .
S ( z ) = [ S 0 ( z ) , S 1 ( z ) , S 2 ( z ) , S 3 ( z ) ] .
s 1 ( z ) = S 1 ( z ) / S 0 ( z ) , s 2 ( z ) = S 2 ( z ) / S 0 ( z ) , s 3 ( z ) = S 3 ( z ) / S 0 ( z ) ,
s = [ s 1 , s 2 , s 3 ]
s = ( s 1 2 + s 2 2 + s 3 2 ) 1 / 2 ,
S ( z ) = M z S ( 0 ) .
d S ( z ) / d z = ( d M z / d z ) S ( 0 ) .
mS ( z ) = ( d M z / d z ) S ( 0 ) ,
mM z S ( 0 ) = ( d M z / d z ) S ( 0 ) ,
d M z / d z = mM z ,
m = ( d M z / d z ) M z - 1 .
R ( - θ ) mR ( θ ) = lim Δ z 0 R ( - θ ) M z , Δ z R ( θ ) - I Δ z ,
m = R ( - θ ) mR ( θ ) ,
m = lim Δ z 0 M z , Δ z - I Δ z ,
M z , Δ z = R ( - θ ) M z , Δ z R ( θ ) ,
R ( θ ) = ( 1 0 0 0 0 cos 2 θ sin 2 θ 0 0 - sin 2 θ cos 2 θ 0 0 0 0 1 ) .
M z = ( 1 0 0 0 0 1 0 0 0 0 cos η z sin η z 0 0 - sin η z cos η z ) ,
η = 4 π Δ n / λ .
M Δ z = ( 1 0 0 0 0 1 0 0 0 0 1 η Δ z 0 0 - η Δ z 1 ) .
m = ( 0 0 0 0 0 0 0 0 0 0 0 η 0 0 - η 0 ) .
J = ( J 1 J 3 J 4 J 2 ) ,
M = ( ( E 1 + E 2 + E 3 + E 4 ) / 2 ( E 1 - E 2 - E 3 + E 4 ) / 2 F 13 + F 42 - D 13 - D 42 ( E 1 - E 2 + E 3 - E 4 ) / 2 ( E 1 + E 2 - E 3 - E 4 ) / 2 F 13 - F 42 - D 13 + D 42 F 14 + F 32 F 14 - F 32 F 12 + F 34 - D 12 + D 34 D 14 + D 32 D 14 - D 32 D 12 + D 34 F 12 - F 34 ) ,
E i = J i 2 ,             F i k = Re ( J i J k * ) ,             D i k = Im ( J i * J k ) ,             i , k = 1 , , 4.
J = I + N Δ z ,
N = ( n 1 n 3 n 4 n 2 ) ,             n k = n k r + i n k i ,             k = 1 , , 4
J = ( 1 + n 1 Δ z n 3 Δ z n 4 Δ z 1 + n 2 Δ z ) .
M = I + m Δ z ,
m = ( n 1 r + n 2 r n 1 r - n 2 r n 3 r + n 4 r - n 3 i + n 4 i n 1 r - n 2 r n 1 r + n 2 r n 3 r - n 4 r - n 3 i - n 4 i n 3 r + n 4 r - n 3 r + n 4 r n 1 r + n 2 r n 1 i - n 2 i - n 3 i + n 4 i n 3 i + n 4 i - n 1 i + n 2 i n 1 r + n 2 r ) .
d + = n 1 + n 2 , d - = n 1 - n 2 , f + = n 3 + n 4 , f - = n 3 - n 4 ,
m = ( d + r d - r f + r - f - i d - r d + r f - r - f + i f + r - f - r d + r d - i - f - i f + i - d - i d + r ) .
m = ( m 1 m 3 m 4 m 2 ) ,
m = ( α β γ δ β α μ ν γ - μ α η δ - ν - η α ) .
m = m s + m s s ,
m s = ( α β γ δ β α 0 0 γ 0 α 0 δ 0 0 α ) ,
m s s = ( 0 0 0 0 0 0 μ ν 0 - μ 0 η 0 - ν - η 0 ) .
M = exp ( m z ) = I + m z + m 2 ( z 2 / 2 ! ) + m 3 ( z 3 / 3 ! ) + ,
m = Tm D T - 1 ,
m D = ( v 1 0 0 0 0 v 2 0 0 0 0 v 3 0 0 0 0 v 4 )
det ( m - v I ) = 0 ,
T = [ S 1 S 2 S 3 S 4 ] ,
mS i = v i S i .
m n = Tm D n T - 1 ,
M = T [ I + m D z + m D 2 ( z 2 / 2 ! ) + ] T - 1 ,
M = TM D T - 1 ,
M D = exp ( m D z ) = I + m D z + m D 2 ( z 2 / 2 ! ) + .
M D = ( e v 1 z 0 0 0 0 e v 2 z 0 0 0 0 e v 3 z 0 0 0 0 e v 4 z )
M = [ S 1 S 2 S 3 S 4 ] ( e v 1 z 0 0 0 0 e v 2 z 0 0 0 0 e v 3 z 0 0 0 0 e v 4 z ) [ S 1 S 2 S 3 S 4 ] - 1 .
( α - v ) 2 [ ( α - v ) 2 - τ 2 ] = 0 ,
τ 2 = β 2 + γ 2 + δ 2 ,
v 1 = v 2 = α ,             v 3 = α + τ ,             v 4 = α - τ .
T = ( 0 0 1 1 0 - ( γ 2 + δ 2 ) 1 / 2 / τ β / τ 2 - β / τ 2 δ / ( δ 2 + γ 2 ) 1 / 2 - β γ / τ ( γ 2 + δ 2 ) 1 / 2 γ / τ 2 - γ / τ 2 - γ / ( δ 2 + γ 2 ) 1 / 2 β δ / τ ( γ 2 + δ 2 ) 1 / 2 δ / τ 2 - δ / τ 2 )
S i t S i = 1 ,             S i t S j = 0 ,             i , j = 1 , , 4
M z = e α z ( cosh τ z β n sinh τ z γ n sinh τ z δ n sinh τ z β n sinh τ z 1 + β n 2 ( cosh τ z - 1 ) β n γ n ( cosh τ z - 1 ) β n δ n ( cosh τ z - 1 ) γ n sinh τ z β n γ n ( cosh τ z - 1 ) 1 + γ n 2 ( cosh τ z - 1 ) γ n δ n ( cosh τ z - 1 ) δ n sinh τ z β n δ n ( cosh τ z - 1 ) γ n δ n ( cosh τ z - 1 ) 1 + δ n 2 ( cosh τ z - 1 ) ) ,
β n = β / τ ,             γ n = γ / τ ,             δ n = δ / τ ;             β n 2 + γ n 2 + δ n 2 = 1.
S ( z + d ) = M z S ( d ) ,
S ( z ) = e α z ( cosh τ z β n sinh τ z γ n sinh τ z δ n sinh τ z ) .
s = ( β n , γ n , δ n ) tanh τ z ,
v 2 [ v 2 + ω 2 ] = 0 ,
ω 2 = μ 2 + ν 2 + η 2 ,
v 1 = v 2 = 0 ,             v 3 = i ω ,             v 4 = - i ω .
S = S 0 ( 1 η c - ν c μ c ) ,
S 1 = ( 1 η n - ν n μ n ) ,             S 2 = ( 1 - η n ν n - μ n ) ,
η n = η / ω ,             ν n = ν / ω ,             μ n = μ / ω ;             η n 2 + ν n 2 + μ n 2 = 1.
S 3 = ( 0 η n 2 - 1 - η n ν n - i μ n μ n η n - i ν n ) ,             S 4 = ( 0 η n 2 - 1 - η n ν n + i μ n μ n η n + i ν n ) .
T = ( 1 1 0 0 η n - η n η n 2 - 1 η n 2 - 1 - ν n ν n - η n ν n - i μ n - η n ν n + i μ n μ n - μ n μ n η n - i ν n μ n η n + i ν n ) .
T - 1 = 1 2 ( 1 η n - ν n μ n 1 - η n ν n - μ n 0 - 1 - η n ν n + i μ n μ n 2 + ν n 2 η n μ n + i ν n μ n 2 + ν n 2 0 - 1 - η n ν n - i μ n μ n 2 + ν n 2 η n μ n - i ν n μ n 2 + ν n 2 ) .
M z = ( 1 0 0 0 0 cos ω z + η n 2 ( 1 - cos ω z ) μ n sin ω z - ν n η n ( 1 - cos ω z ) ν n sin ω z + μ n η n ( 1 - cos ω z ) 0 - μ n sin ω z - ν n η n ( 1 - cos ω z ) cos ω z + ν n 2 ( 1 - cos ω z ) η n sin ω z - μ n ν n ( 1 - cos ω z ) 0 - ν n sin ω z + μ n η n ( 1 - cos ω z ) - η n sin ω z - μ n ν n ( 1 - cos ω z ) cos ω z + μ n 2 ( 1 - cos ω z ) ) .
S 0 = 1 , S 1 = μ n η n ( 1 - cos ω z ) + ν n sin ω z , S 2 = - μ n ν n ( 1 - cos ω z ) + η n sin ω z , S 3 = μ n 2 ( 1 - cos ω z ) + cos ω z .
η n S 1 - ν n S 2 + μ n S 3 = μ n ,
( α - v ) 4 + ( α - v ) 2 [ ( μ 2 + ν 2 + η 2 ) - ( β 2 + γ 2 + δ 2 ) ] + 2 [ β γ ν η - β δ μ η + γ δ μ ν ] = 0.
m = R ( - ρ z ) m 0 R ( ρ z ) ,
ρ = 2 π / p ,
m = ( 0 0 0 0 0 0 0 η sin 2 ρ z 0 0 0 η cos 2 ρ z 0 - η sin 2 ρ z - η cos 2 ρ z 0 ) ,
d S 0 / d z = 0 ,
d S 1 / d z = ( η sin 2 ρ z ) S 3 ,
d S 2 / d z = ( η cos 2 ρ z ) S 3 ,
d S 3 / d z = ( - η sin 2 ρ z ) S 1 + ( - η cos 2 ρ z ) S 2 .
( d 2 S 3 / d z 2 ) + η 2 S 3 = 2 η ρ ( - S 1 cos 2 ρ z + S 2 sin 2 ρ z ) .
- S 1 = ( sin 2 ρ z / η ) ( d S 3 / d z ) + ( cos 2 ρ z / 2 η ρ ) ( d 2 S 3 / d z 2 ) + ( η cos 2 ρ z / 2 ρ ) S 3 .
( d S 3 3 / d z 3 ) + ( η 2 + 4 ρ 2 ) ( d S 3 / d z ) = 0.
S 3 = c 1 + c 2 sin ( ζ z + c 3 ) , ζ 2 = η 2 + 4 ρ 2 ,
S 1 = c 4 + ( - η c 1 / 2 ρ ) cos 2 ρ z + [ η c 2 / 2 ( 2 ρ - ζ ) ] sin [ ( 2 ρ - ζ ) z - c 3 ] + [ - η c 2 / 2 ( 2 ρ + ζ ) ] sin [ ( 2 ρ + ζ ) z + c 3 ] ,
S 2 = c 5 + ( η c 1 / 2 ρ ) sin 2 ρ z + [ - η c 2 / 2 ( 2 ρ + ζ ) ] cos [ ( 2 ρ + ζ ) z + c 3 ] + [ η c 2 / 2 ( 2 ρ - ζ ) ] cos [ ( 2 ρ - ζ ) z - c 3 ] ,
c 1 = ( 4 ρ 2 / ζ 2 ) S 30 + ( - 2 η ρ / ζ 2 ) S 10 ,
c 2 sin c 3 = ( η 2 / ζ 2 ) S 30 + ( 2 η ρ / ζ 2 ) S 10 ,
c 2 cos c 3 = ( - η / ζ ) S 20 .
M 11 = 1 ,             M 12 = M 13 = M 14 = M 21 = M 31 = M 41 = 0 ; M 22 = ( ρ η 2 / ζ 2 ) ( cos 2 ρ z ρ - cos ( 2 ρ + ζ ) z 2 ρ + ζ - cos ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 23 = ( η 2 / 2 ζ ) ( sin ( 2 ρ + ζ ) z 2 ρ + ζ - sin ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 24 = ( - η / 2 ζ 2 ) × ( 4 ρ cos 2 ρ z + η 2 cos ( 2 ρ + ζ ) z 2 ρ + ζ + η 2 cos ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 32 = ( ρ η 2 / ζ 2 ) ( - sin 2 ρ z ρ + sin ( 2 ρ + ζ ) z 2 ρ + ζ + sin ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 33 = ( η 2 / 2 ζ ) ( cos ( 2 ρ + ζ ) z 2 ρ + ζ - cos ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 34 = ( η / 2 ζ 2 ) ( 4 ρ sin 2 ρ z + η 2 sin ( 2 ρ + ζ ) z 2 ρ + ζ + η 2 sin ( 2 ρ - ζ ) z 2 ρ - ζ ) , M 42 = ( - 2 ρ η / ζ 2 ) ( 1 - cos ζ z ) ,             M 43 = ( - η / ζ ) sin ζ z , M 44 = ( 4 ρ 2 + η 2 cos ζ z ) / ζ 2 .
m = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - σ ) ,
d S 0 / d z = d S 1 / d z = d S 2 / d z = 0 ,
d S 3 / d z = - σ S 3 .
S 3 = S 30 e - σ z .