Evolution of partially polarized light through non-depolarizing anisotropic media

1 January 2000

Optics Communications 173 Ž2000. 57–71 www.elsevier.comrlocateroptcom

Evolution of partially polarized light through non-depolarizing anisotropic media J.F. Mosino, ˜ O. Barbosa-Garcıa ´ ) , A. Starodumov, L.A. Dıaz-Torres, ´ M.A. Meneses-Nava, Jose´ T. Vega-Duran ´ Centro de InÕestigaciones en Optica, A.P. 1-948, Leon, ´ Gto. 37000 Mexico Received 31 August 1999; accepted 26 October 1999

Abstract An analytical solution for the propagation equation of partially polarized light trough anisotropic media is reported. Our solution gives an explicit expression for the Mueller matrix of uniform non-depolarizing optical media. The state of polarization of quasi-monochromatic partially polarized light is expressed by the Stokes vector. The medium is assumed to be homogeneous, linear and characterized by a differential Mueller matrix with arbitrary and combined effects of absorption and dispersion. The reported solution gives the general behavior of the state of polarization of light through any media with arbitrary and combined deterministic values of dichroism and birefringence, which has not been reported in the literature. The particular cases of media with pure dichroism or pure birefringence are also included in our formalism. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.81.G; 42.25.J; 78.20.F Keywords: Mueller matrix; Birefringence; Dichroism; Polarization; Stokes vector

1. Introduction It is well known that as light propagates in anisotropic media, its state of polarization changes. Thus, new studies are carried on to deeper understand the evolution of the state of polarization of light traveling through different media. Such studies have defined a growing field in both, theoretical and experimental optics, because of their practical applications. The characterization of the evolution of the state of polarization is of great importance for the development of devices such as polarimetric sensors, liquid crystals displays, medical instruments, fiber lasers and amplifiers and fiber optic communications w1–5x. For studying the propagation of monochromatic and totally polarized light through homogeneous, linear and non-depolarizing media, the Jones vector and the coherence matrix formalisms can be used w6x. That is, the evolution of the state of polarization along the propagation path is easily characterized by such formalisms.

)

Corresponding author. Tel.: q52-4-717-5823; fax: q52-4-717-5000; e-mail: [email protected]

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 4 7 - 1

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However, for partially polarized light or depolarizing media, the Jones formalism cannot longer be applied and the Stokes vector and Mueller matrix formalisms w7–9x should be used instead. In the Stokes vector and Mueller matrix formalisms, the evolution of the state of polarization of light as it propagates in the z-direction is described by the coupled first order non-linear differential equations d Srd z s mS

Ž 1.

where S is the 4 = 1 Stokes vector and m is a 4 = 4 real matrix that summarizes the optical intensive properties of the medium. The matrix m is also known as the differential Mueller matrix w7x. For depolarizing media, the differential Mueller matrix is defined by sixteen independent parameters. However, for uniform, linear and non-depolarizing media m can be defined by only seven independent parameters w7x. Such deterministic parameters account for arbitrary and combined dichroisms and birefringences, i.e., four independent parameters to describe the isotropic and anisotropic absorption properties of the medium and three more for the anisotropic refraction properties of the medium. The explicit form for matrix m is given by w7x

a b ms g d

b a ym yn

g m a yh

d n h a

Ž 2.

where a is the isotropic absorption, b is the linear dichroism along the x–y coordinate axes, where x–y–z form an orthogonal coordinate system fixed to the medium, g is the linear dichroism along the bisectors of x–y coordinate axes, d is the circular dichroism, h is the linear birefringence along the x–y coordinate axes, n is the linear birefringence along the bisectors of x–y coordinate axes, and m is the circular birefringence. Therefore, from Eq. Ž1. the solution for the propagation of the Stokes vector SŽ z . with an initial condition Ž S 0. at z s 0, is given by S Ž z . s exp Ž mz . S Ž 0 . s MS Ž 0 .

Ž 3.

The exponential of mz in Eq. Ž3. defines the so-called Mueller matrix M or matrix of extensive properties of the medium w7x. Thus, for ideal media with only arbitrary absorptive or only arbitrary refractive anisotropies Azzam w7x obtained the analytical form of the Mueller matrix. On the other hand, Brown and Bak w1x developed a formalism based on the Lorentz group theory to obtain analytical expressions for the Mueller matrix for the same particular cases reported by Azzam w7x. However, these two research groups failed to report an analytical and explicit expression for the case of a medium with combined values of birefringence and dichroism when each of them has nonzero values. Therefore, with the reported solutions, a medium with non zero values of net dichroism and net birefringence has to be modeled by a system with pure dichroism or pure birefringence. This kind of approximation is widely in use because, to our knowledge, no analytical and explicit solution has been reported for the combined effects of dichroism and birefringence. It should be noticed that the solution reported by Brown and co-workers can only be applied numerically. However, the physical insights are not clear because the solution is reported in terms of the Lorentz group formalism. An analytical and explicit expression of the Mueller matrix, for such media is needed to calculate and get the insights of the evolution of the state of polarization of light. In this paper, we report an analytical and explicit solution for the general case of non-depolarizing media where both, the birefringence and dichroism have non-zero values. This solution is obtained from Eq. Ž3. with the Sylvester’s interpolation method w10x. Our solution shows that any behavior of the evolution of the Stokes vector can be determined through six general cases. Some of these cases have been reported experimentally w1–5x. As expected, it is shown that any case reported in the literature falls within one of the six cases obtained from our solution.

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2. The Mueller matrix The first step to calculate the Mueller matrix is to calculate the eigenvalues w10x of the differential Mueller matrix m. These eigenvalues are given by the roots of the characteristic equation det < m y l I < s 0, where I is the 4 = 4 identity matrix and m is defined by Eq. Ž2.. Solving such determinant, the characteristic equation is given by 4 2 2 Ž a y l . y Ž a y g . Ž b 2 q g 2 q d 2 . y Ž h 2 q m2 q n 2 . y Ž bh y ng q md . s 0.

Ž 4.

Solving for the roots of Eq. Ž4., the eigenvalues of m are

l1 s a y t , l2 s a q t , l 3 s a y i V , l4 s a q i V ,

Ž 5.

where t G 0 and V G 0 are given by

° Žb ¢ž

t s~

2

1r2

2

q g 2 q d 2 . y Ž h 2 q m2 q n 2 .

/

2

q Ž bh y ng q md .

Ž b 2 q g 2 q d 2 . y Ž h 2 q m2 q n 2 .¶ •

2

1r2

q

° Žb V s~ ¢ž

2

.

ß

2

Ž 6. 1r2

2

q g 2 q d 2 . y Ž h 2 q m2 q n 2 .

/

2

Ž b 2 q g 2 q d 2 . y Ž h 2 q m2 q n 2 .¶ •

q Ž bh y ng q md .

2

1r2

y

2

ß

.

Ž 7.

Now, we introduce two three-dimensional vectors defined as x s  b ,g , d 4 and y s h ,y n , m4 . In Refs. w1,7x, equivalent vectors were introduced and their norms represent the net dichroism and net birefringence of the medium. Thus, we refer to x and y as the dichroism and birefringence vectors, respectively. Notice that for any given differential Mueller matrix, Eq. Ž2., there exist two real vectors x and y. Therefore, the scalar magnitudes of t and V correspond to physical properties of the medium and we called them the effective dichroism and effective birefringence, respectively. Then, the product between t and V , is given by the absolute value of the dot product between these vectors, that is, < x P y < s tV s < bh y ng q md < .

Ž 8.

Furthermore, the difference of the square norms of these two vectors is given by < x < 2 y < y < 2 s t 2 y V 2 s Ž b 2 q g 2 q d 2 . y Ž h 2 q n 2 q m2 . .

Ž 9.

It should be pointed out that the dot product between x and y defines what we call the coupling between the dichroism and birefringence vectors. That is, the normalized coupling between the x and y vectors is given by cos Ž u . s

xPy < x < < y<

,

Ž 10 .

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where u corresponds to the coupling angle. Therefore, we can express Eqs. Ž6. and Ž7. as < x < 2 y< y< 2

ž

2

t s

2

V s

2

ž

< x < 2 y< y< 2 2

1r2

2

/

q Ž x Py.

2

q

ž

2

1r2

2

/

< x <2 y< y<2

q Ž x Py.

2

y

ž

< x < 2 y< y< 2 2

/

Ž 11 .

/

.

Ž 12 .

When the eigenvalues given by Eq. Ž5. are expressed through Eqs. Ž11. and Ž12., they clearly show a dependency on x P y and < x < 2 y < y < 2 . By introducing the x and y vectors and the coupling given by Eq. Ž10., one clearly identifies six general cases for the behavior of Stokes vector as discussed below in Section 3. The Sylvester’s interpolation method w10x is a simple and systematic procedure to obtain an analytical and explicit form of the exponential of a matrix. The Sylvester’s determinant to solve the Mueller matrix, M s expŽ mz ., takes the following form, 1



1

det 1 1 I

l1

l12

l13

exp Ž l1 z .

l2

l22

l32

exp Ž l 2 z .

l3

l23

l33

exp Ž l3 z . s 0.

l4

l24

l34

exp Ž l4 z .

m

2

3

m

m

exp Ž mz .

0

Ž 13 .

Expanding Eq. Ž13. by Cramer’s rule w10x, we obtain, exp Ž mz . s Ž 1rs 0 . ys 1exp Ž l1 z . q s 2 exp Ž l2 z . y s 3 exp Ž l 3 z . q s4 exp Ž l4 z . ,

Ž 14 .

where the si are defined by the following Vandermonde’s determinants w10x

s 0 s Ž l4 y l3 . Ž l4 y l2 . Ž l4 y l1 . Ž l3 y l2 . Ž l3 y l1 . Ž l2 y l1 . s 1 s Ž l4 y l3 . Ž l4 y l2 . Ž l3 y l2 . Ž m y l4 I . Ž m y l3 I . Ž m y l2 I . s 2 s Ž l4 y l3 . Ž l4 y l1 . Ž l3 y l1 . Ž m y l4 I . Ž m y l3 I . Ž m y l1 I . s 3 s Ž l4 y l2 . Ž l4 y l1 . Ž l2 y l1 . Ž m y l4 I . Ž m y l2 I . Ž m y l1 I . s4 s Ž l3 y l2 . Ž l3 y l1 . Ž l2 y l1 . Ž m y l3 I . Ž m y l2 I . Ž m y l1 I . .

Ž 15 .

At this point, we introduce the matrix of anisotropies defined as C s m y a I, thus, M becomes, Ms

ŽCyV i I . ŽCqV i I . ŽCyt I . ŽCyV i I . ŽCqV i I . ŽCqt I . exp Ž l1 z . q exp Ž l 2 z . Ž l1 y l4 . Ž l1 y l3 . Ž l1 y l2 . Ž l2 y l4 . Ž l2 y l3 . Ž l2 y l1 . q

ŽCyt I . ŽCqt I . ŽCyV i I . ŽCyt I . ŽCqt I . Ž CqV i I . exp Ž l3 z . q exp Ž l4 z . . Ž l3 y l4 . Ž l3 y l2 . Ž l3 y l1 . Ž l4 y l3 . Ž l4 y l2 . Ž l4 y l1 .

Ž 16 .

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Using the Cayley–Hamilton theorem w10x, and the expressions for the matrix of anisotropies C and its inverse Cy1 , the cubic power C 3 is given by 0 b Cs g d

b 0 ym yn

g m 0 yh

d 0 1 n h , Cy1 s h y n xPy 0 m

h 0 d yg

yn yd 0 b

m g , C 3 s Ž t 2 y V 2 . C q t 2V 2 Cy1 . yb 0

Ž 17 . Therefore, using Eq. Ž17. and the eigenvalues given by Eq. Ž5. into Eq. Ž16., we get the following compact result for the Mueller matrix M Ms

exp Ž a z .

t 2qV 2



V 2 cosh Ž t z . q t 2 cos Ž V z . I q cosh Ž t z . y cos Ž V z . C 2

q t sinh Ž t z . q V sin Ž V z . C q Ž tV . V sinh Ž t z . y t sin Ž V z . Cy1 4 .

Ž 18 .

When we use Eq. Ž18. into Eq. Ž3., the analytical and explicit solution to the differential Eq. Ž1. is obtained. Thus, the Stokes vector can be evaluated at any distance when light propagates through any media with arbitrary values of anisotropies, including nonzero values for each anisotropy, i.e., dichroism and birefringence. Furthermore, Eq. Ž18. can be explicitly rewritten in terms of the anisotropies defined by Eq. Ž2. and this expression is reported in Appendix A.

3. Discussion Let us make some remarks about Eq. Ž18.. The isotropic absorption parameter a is an exponential factor, expŽ a z ., that does not affect the state of polarization of the light. Notice that Eq. Ž18. is reported by factoring out I, C, C 2 and inverse Cy1 , but it could also be reported by grouping the hyperbolic and non-hyperbolic functions. Thus, the state of polarization depends on the sum of an oscillatory part and a hyperbolic part. The oscillatory part is a function of the V parameter that represents the frequency, in units of radrm, of the oscillations of the Stokes vector. So, we call this parameter the effective birefringence of the medium, with V s 2prT and, it can be measured from the period T of the oscillations of any of the Stokes parameters. On the other hand, the hyperbolic part is a function of the t parameter, in units of radrm, that is, t acts as saturation factor. This parameter is also measurable and so, we call it the effective dichroism of the medium. Furthermore, we can remark that by putting Eqs. Ž11. and Ž12. into Eq. Ž5., the eigenvalues depend on the dot product and on the difference of the square norms of the dichroism and birefringence vectors. From this

Table 1 Zero coupling xP ys 0 Case I

Case II

Case III

< x < s < y<

< x < - < y<

< x < ) < y<

t s0

t s0

t s < x < 2 y < y< 2

V s0 Polynomial behavior Reported in literature for < x < s < y < s 0 as Lambert’s law

V s < y< 2 y < x < 2 Sinusoidal behavior Reported in literature for < y < s 0 as pure birefringence

(

(

V s0 Hyperbolic behavior Reported in literature for < x < s 0 as pure dichroism

J.F. Mosino ˜ et al.r Optics Communications 173 (2000) 57–71

62 Table 2 No zero coupling xP y/ 0 Case IV

Case V

Case VI

< x < s < y< t sV

< x < - < y< t -V Eqs. Ž11. and Ž12. for t and V HyperbolicqSinusoidal Behavior Not reported in literature

< x < ) < y< t )V Eqs. Ž11. and Ž12. for t and V Hyperbolic behavior Not reported in literature

'

V st s < xP y < Hyperbolicqsinusoidal behavior Not reported in literature

observation, we distinguish two cases according to the zero or nonzero value of the dot product, i.e., zero coupling and non-zero coupling between such vectors. For each of these two cases one can define three particular cases according to the magnitudes of the dichroism and birefringence vectors, i.e., < x < s < y <, < x < - < y < and < x < ) < y <. These six cases give the behavior of the Stokes vector as it propagates through any media, which is characterized by Eq. Ž2.. Tables 1 and 2 show the main features for coupling and no-coupling between the x and y vectors, respectively. Let us now analyze any of the six cases. The first three cases correspond to zero coupling, this means that the dichroism and birefringence vectors are orthogonal. 3.1. Case I. No coupling, x P y s 0, and < x < s < y < For this case, we observe from Eqs. Ž11. and Ž12. that V s 0 and t s 0. Thus, from Eq. Ž5. each eigenvalue, l1 , l2, l3 and l4 , is equal to a . Thus, using these values in Eq. Ž18. and putting the result in Eq. Ž3. we obtain

ž

S Ž z . s exp Ž a z . I q C q

1 2!

C 2 S Ž 0. .

/

Ž 19 .

To obtain Eq. Ž19., one should take one limit at a time in Eq. Ž18., otherwise an undetermined form Ž0r0. is obtained. The explicit expression of the Mueller matrix M can be obtained from Appendix A, and the result is M sexp Ž a z .

°

1q Ž b 2 qg 2 q d 2 .

b z q Ž gm q dn .

= g z q Ž y bm q dh .

¢d

z q Ž y bn ygh .

z2 2 z2 2 z2 2 z

b z q Ž ygm y dn . 1q Ž b 2 y m2 y n . y m z Ž bg y nh .

2

2

y n z q Ž bd qhm .

z2 2 z2 2 z2 2 z

g z q Ž bm y dh . m z q Ž bg y nh . 1q Ž g 2 y m2 yh 2 .

2

2

yh z q Ž gd y nm .

z2 2 z2 2 z2 2 z

¶

d z q Ž bn qgh . n z q Ž bd qhm . h z q Ž gd y nm .

2

2

1q Ž d 2 y n 2 yh 2 .

z2 2

z2 2 z2

.

Ž 20 .

2 z2

ß

2

We evaluate the Stokes vector behavior by using Eq. Ž20. into Eq. Ž3.. To do our calculations, we assumed an incident natural light, i.e., SŽ0. s  1,0,0,04 . The dichroism and birefringence vectors of the medium were chosen as x s  4,5,34 and y s  4,y 5,34 . In Fig. 1 we report the calculations for the normalized Stokes parameter S3rS0 as a dashed curve. At very short distances, the dashed curve shows a maximum and at larger distances, it asymptotically approaches some fixed value. On the other hand, the degree of polarization w7x, DoP s

(S q S q S 2 1

2 2

S0

2 3

,

Ž 21 .

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Fig. 1. Case I–II. Spatial evolution of S3 rS0 for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP ys 0. All curves, solid and dashed curves, correspond to the following ratios between net dichroism and net birefringence, r s 0, 0.05, 0.55, 0.97 for solid curves a–d, and r s1 for the dashed curve. The magnitude of the net birefringence is '50 my1 for the vector ys 4,y5,34 while x s r4,5,34.

shows a behavior similar to a hyperbolic tangent with an asymptotic behavior to unity as shown in Fig. 2 in the dashed curve. This result implies that the incident natural light becomes totally polarized after a long propagation distance with the net dichroism < x < as the saturation factor. This confirms the non-depolarizing nature of the medium. For a particular medium with null dichroism and null birefringence, < x < s < y < s 0, Eq. Ž20. becomes the identity matrix multiplied by the exponential of the isotropic absorption factor. This result represents the Lambert’s law that holds for media with isotropic absorption and isotropic refraction, and with a null matrix of anisotropies C s 0.

Fig. 2. Case I–II. Spatial evolution of the degree of polarization ŽDoP. for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP ys 0. The solid and dashed curves correspond to the same values of r, x and y of the curves in Fig. 1.

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3.2. Case II. No coupling, x P y s 0, and < x < - < y < From Eq. Ž11. we can observe that the second term is negative with the same magnitude than the first term, thus, t s 0; meanwhile, from Eq. Ž12., we have V s < y < 2 y < x < 2 . Using these two results into Eq. Ž5., we get l1 s l2 s a , l3 s a y i V and l4 s a q i V . Putting these eigenvalues into Eqs. Ž18. and Ž3., we get the Stokes vector

(

SŽ z . s

exp Ž a z .

V2

 V 2 I q V sin Ž V z . C q

1 y cos Ž V z . C 2 4 S Ž 0 . .

Ž 22 .

The explicit form for the Mueller matrix can be obtained using the general expression reported in Appendix A. We evaluate Eq. Ž22. for SŽ0. s  1,0,0,04 , x s r 4,5,34 and y s  4,y 5,34 , where r is defined as r s < x
(

From the Eq. Ž12. we can see that V s 0 and, from Eq. Ž11. we have t s < x < 2 y < y < 2 . Using these two results into Eq. Ž5., we obtain, l1 s a y t , l1 s a q t and l3 s l 4 s a . Using these values into Eqs. Ž18. and Ž3., we get the following Stokes vector, SŽ z . s

exp Ž a z .

t2

 t 2 I q t Csinh Ž t z . q C 2 Ž cosh Ž t z . y 1. 4 S Ž 0. .

Ž 23 .

The explicit form for the Mueller matrix can be obtained using the general expression reported in Appendix A with V s 0. We evaluate Eq. Ž23. for SŽ0. s  1,0,0,04 , x s  4,5,34 and y s r 4,y 5,34 , where r is defined as 0 - r - 1. Notice that x and y are orthogonal vectors while the allowed values for r defines the condition < x < ) < y <. Thus, for r s 0, 0.2, 0.35 and 0.8 we have plotted the solid curves a, b, c and d, respectively, in Fig. 3 for S3rS0 . Notice that the dashed curve in Fig. 3 corresponds to the case I discussed above when the ratio r approaches to the unity value. As in the previous case, this solution also describes a hyperbolic evolution of the Stokes vector with a local maximum at the beginning of the propagation, and an asymptotic behavior at longer distances. Notice that the oscillation amplitude in Fig. 3 vanishes as the ratio r of net birefringence to net dichroism decreases, therefore,

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Fig. 3. Case III. Spatial evolution of S3 rS0 for natural light incident on non-depolarizing media with simultaneous effects of absorption and dispersion, for xP ys 0. All curves, solid and dashed curves, correspond to the following ratios between net birefringence and net dichroism r s 0, 0.2, 0.35 and 0.8 for solid curves a–d, and r s1 for the dashed curve. The magnitude of the net dichroism is '50 my1 for the vector x s 4,5,3. while ys r4,y5,34.

for r s 0, no oscillation is observed in curve a. Furthermore, as in case II, if the net birefringence is an order of magnitude smaller than the net dichroism, the parameter of saturation t becomes t ( < x <, within an error smaller than 0.5%. This result is also an implicit assumption often made to interpret some experimental results as in Ref. w1x. As expected, Eq. Ž23. is equivalent to the solution reported in the literature w1,7x for the media with pure dichroism, i.e., < y < s 0. On the other hand, the behavior of the degree of polarization is shown in Fig. 4, solid curves a–d. Notice that there are four solid curves and one dashed in Fig. 4 that are very similar one to the other. For the dashed curve, r s 1 as in case I discussed above. This ratio r decreases down to the zero value as reported above and corresponds to the upper curve. That is, the evolution for the degree of polarization for this case is very similar to a hyperbolic tangent of t z. To observe in detail the differences among these curves, we have plotted in the inset an enlarged view.

Fig. 4. Case III. Spatial evolution of the degree of polarization ŽDoP. for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP ys 0. The solid and dashed curves correspond to the same cases of the curves in Fig. 3.

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Fig. 5. Case IV. Spatial evolution of S3 rS0 for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP y/ 0. All solid curves, correspond to the following ratio r s1 between net dichroism and net birefringence for coupling values cosŽ u . s6=10y5 , 0.14, 0.42, 0.7 and 1 for curves a–e respectively. The magnitude of the net birefringence and net dichroism is '50 my1 for the fixed vector ys 4,y5,34 with x s 4,5,3.0014, 3,4,54, 0,0,'50 4, 5,0,54 and 4,y5,34, for the curves a–e respectively.

Let us now consider the other three cases, when the x and y vectors are not orthogonal and the coupling takes non-zero values, i.e., x P y / 0. We make use of Eq. Ž10. to quantify the coupling. For these cases, no general simplification can be made, and the eigenvalues are defined by Eq. Ž5. and the general explicit solution is always given by the expression reported in Appendix A. 3.4. Case IV. Coupling, x P y / 0 and < x < s < y < / 0

(

(

From Eqs. Ž11. and Ž12., we can observe that t s V s < x P y < s < y < cos Ž u . . We evaluated the Stokes vector according to Eqs. Ž18. and Ž3.. To our calculations, we consider the incident Stokes vector SŽ0. s  1,0,0,04 , and y s  4,y 5,34 , while x takes the following values x s  4,5,3.0014 ,  3,4,54 ,  0,0,'50 4 ,  5,0,54 and  4,y 5,34 , for the curves a–e respectively. Thus, the plot of S3rS0 against distance is shown in Fig. 5 for cosŽ u . s 6 = 10y5 , 0.14, 0.42, 0.7 and 1, that correspond to curves a–e, respectively. Notice that for the 6 = 10y5 value of the coupling, the behavior is similar to case I. Furthermore, Eq. Ž10. allows negative values for the coupling, and Fig. 6 shows the behavior resulting from using the negative values of the coupling used for Fig. 5. In Fig. 6 we observed that the evolution of the Stokes vector has a change of phase of 1808 compared to Fig. 5. For both cases in Figs. 5 and 6 when the absolute coupling value increases to unity, the value of t decreases the saturation length. For the evolution of the degree of polarization, see Fig. 7, the shape is similar to case III in Fig. 4. 3.5. Case V. Coupling, x P y / 0 and < x < - < y < To evaluate the behavior of the Stokes vector in this case, and using Eq. Ž10., one can rewrite Eqs. Ž11. and Ž12. as, < y< 2

2

Ž r y 1 . q 2 rcos Ž u . 2 ½ < y< V s Ž r y 1 . q 2 rcos Ž u . 2 ½

t 2s

2

1r2

2

2

2

2

2

2

5 q1yr 5 .

y1qr2

1r2

2

Ž 24 . Ž 25 .

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Fig. 6. Case IV. Spatial evolution of S3 rS0 for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP y/ 0. All solid curves, correspond to the following ratio r s1 between net dichroism and net birefringence for coupling values cosŽ u . sy6=10y5 , y0.14, y0.42, y0.7 and y1 for solid curves a–e respectively. The magnitude of the net birefringence and net dichroism is '50 my1 for the fixed vector ys 4,y5,34 while x s y4,y5,y3.0014, y3,y4,y54, 0,0,y'50 4, y5,0,y54 and y4,5,y34, for the curves a–e respectively.

These expressions show the dependency on the coupling value, cosŽ u ., and on r for the effective dichroism and net birefringence. From the conditions of this case t - V and 0 - r - 1. Thus, to our calculations, we consider the incident Stokes vector SŽ0. s  1,0,0,04 , and y s  4,y 5,34 , while x takes the following values x s r 4,5,3.0014 , r 3,4,54 , r 0,0,'50 4 , r 5,0,54 and r 4,y 5,34 with a fixed value of r s 0.5, for the curves a–e respectively. The evaluation of the Stokes vector was carried out through Eqs. Ž18. and Ž3., and the plots of S3rS0 against distance for positive values of the coupling are shown in Fig. 8. Curves a–e corresponds to increasing values of the coupling from 6 = 10y5 , 0.14, 0.42, 0.70 and 1.0, respectively. Thus, for smaller values of the coupling, Fig. 8 shows oscillatory behaviors at any propagation distances, see curves a and b. As the

Fig. 7. Case IV. Spatial evolution of the degree of polarization ŽDoP. for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP ys 0. The solid curves correspond to the same cases of the curves in Figs. 5 and 6. For coupling values cosŽ u . s"6=10y5 , "0.14, "0.42, "0.7 and "1 for curves a–e respectively.

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Fig. 8. Case V. Spatial evolution of S3 rS0 for incident natural light propagating through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP y/ 0. All solid curves, correspond to the following ratio r s 0.5 between net dichroism and net birefringence for coupling values cosŽ u . s6=10y5 , 0.14, 0.42, 0.7 and 1 for solid curves a–e respectively. The magnitude of the net birefringence and net dichroism is 0.5 '50 my1 and '50 my1 respectively for the fixed vector ys 4,y5,34 while x s 0.54,5,3.0014, 0.53,4,54, 0.50,0,'50 4, 0.55,0,54 and 0.54,y5,34, for the curves a–e respectively.

value of the coupling increases and the propagation takes place, see curves c–e, the effects of the net dichroism dominate and the observed oscillatory behavior vanishes. When smaller values of r and small values of the coupling are considered, similar curves to those reported for case II were observed. For r ) 0.5 and large values of the coupling, similar curves to those reported for case IV were observed. These behaviors can easily be observed from Eqs. Ž24. and Ž25.. For negative values of the coupling, the evolution of the Stokes vector showed a change of phase of 1808 compared to Fig. 8. Fig. 9 shows the behavior of the degree of polarization for the considering values of coupling, that is, curves a–e correspond to increasing values of the coupling from 6 = 10y5 , 0.14, 0.42, 0.70 and 1.0. As the coupling increases, the effective dichroism also increases attenuating the amplitude of the oscillations as shown in Fig. 9. For larger values of the coupling, the behavior of the degree of polarization approaches the case III.

Fig. 9. Case V. Spatial evolution of the degree of polarization ŽDoP. for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP y/ 0. The solid curves correspond to the same cases of the curves in Fig. 8. For coupling values cosŽ u . s"6=10y5 , "0.14, "0.42, "0.7 and "1 for curves a–e respectively.

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Fig. 10. Case VI. Spatial evolution of S3 rS0 for natural light incident through linear non-depolarizing media with simultaneous effects of absorption and dispersion, for xP y/ 0. All the different plots correspond to a ratio r s 0.5 between net birefringence and net dichroism but different values of cosŽ u ., which takes the values 6=10y5 , 0.14, 0.42, 0.7 and 1 for solid curves plots a, b, c, d and e respectively. The magnitude of the net birefringence and net dichroism is '50 my1 and 0.5 '50 my1 respectively, for the fixed vector ys 4,y5,34 while x s 0.54,5,3.0014, 0.53,4,54, 0.50,0,'50 4, 0.55,0,54 and 0.54,y5,34, for curves a–e respectively.

3.6. Case VI. Coupling, x P y / 0, with < x < ) < y < For this case, we rewrite Eqs. Ž11. and Ž12. as a function of cosŽ u . and of the ratio ry1 s < y
Žr 2 ½ < x< V s Žr 2 ½

t 2s

y2

2

y 1 . q 2 ry1 cos Ž u .

2

2

y2

2

y 1 . q 2 ry1 cos Ž u .

2 1r2

q 1 y ry2

2 1r2

5 5.

y 1 q ry2

Ž 26 . Ž 27 .

From Eqs. Ž26. and Ž27., we observe that V - t for 0 - ry1 - 1. To evaluate the Stokes vector we assumed SŽ0. s  1,0,0,04 , and y s 0.5 4,y 5,34 , while x takes the following values, x s  4,5,3.0014 ,  3,4,54 ,  0,0,'50 4 ,  5,0,54 and  4,y 5,34 , for the curves a–e respectively. Fig. 10 shows the behavior of S3rS0 against distance, and curves a–e correspond to increasing values of the coupling from cosŽ u . s 6 = 10y5 , 0.14, 0.42, 0.7 and 1, respectively. For the smaller values of the coupling, a similar behavior to case III is found, and for larger values of the coupling, the behavior is similar to case I. These behaviors can be observed from Eqs. Ž26. and Ž27.. Notice that the effective dichroism is greater than the effective birefringence, thus, no oscillatory behavior is observed in Fig. 10. Again, for negative values of the coupling, the evolution of the Stokes vector shows a change of phase of 1808 compared to Fig. 10. The curves for the degree of polarization for this case do not show great differences among the considered coupling values. In general, the behavior of such curves is hyperbolic tangent type similar to those reported for case III, see Fig. 4.

4. General remarks and conclusions In summary, we have presented the explicit analytical solution for the propagation of the state of polarization, given by the Stokes vector, of totally or partially polarized light traveling through anisotropic non-depolarizing media. The medium is characterized by the differential Mueller matrix through seven

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independent parameters. The exact solution was obtained using the Sylvester’s interpolation method and using scalar magnitudes defined by the effective dichroism and effective birefringence. Furthermore, it was shown that for any given differential Mueller matrix, two vectors can be defined as the dichroism and birefringence vectors. A coupling between these two vectors can be defined through their scalar product. From here, we distinguished two cases for the behavior of the Stokes vector, that is, when the coupling is zero, for orthogonal vectors and, when it is non-zero for non-orthogonal vectors. For each case, three additional cases can be considered when the net dichroism is equal, less or greater than the net birefringence, i.e. from the relationship of the magnitude of the vectors. All six cases were discussed in terms of the propagation of the Stokes vector for an incident natural light. The results of our calculations were plotted as S3rS0 against distance of propagation. We also plotted the degree of polarization against distance. It should be pointed out that only three particular cases have been partially reported in the literature. That is, for case I the Lambert’s law can be obtained for the particular case that x s y s 0. For the case II, Azzam and Brown calculated the Mueller matrix for a particular medium with x s 0, and for case III, the same authors calculated the Mueller matrix for y s 0. Notice that our formalism shows that the picture given by Azzam for pure dichroism and pure birefringence is not a general case and it is only valid when one anisotropy is one order of magnitude greater than the other anisotropy. Therefore, to our knowledge, the six general cases reported here have not been reported before and, only three particular cases have been discussed by other authors. From the six cases already discussed, the experimentalist can obtain a qualitative behavior for the Stokes vector as it propagates in any medium that is characterized by the differential Mueller matrix. That is, it does not matter what values for the seven parameters of the Mueller matrix are used, the general behavior of the Stokes vectors can always be obtained from the above discussed cases.

Acknowledgements J.F.M. acknowledges helpful discussions with Dr. L.A. Zenteno. This work was partially supported by the CONACyT, Mexico. ´

Appendix A The analytical and explicit form of the Mueller matrix M s expŽ mz . is given by:

Ms

exp Ž a z .

t 2qV 2

M11 M12 M13 M14

M21 M22 M23 M24

M31 M32 M33 M34

M41 M42 , M43 M44

where M11 s V 2 q b 2 q g 2 q d 2 cosh Ž t z . q t 2 y b 2 y g 2 y d 2 cos Ž V z . M12 s Ž Vb y hht . sin Ž V z . q Ž tb q hhV . sinh Ž t z . q Ž gm q dn . cosh Ž t z . y cos Ž V z . M13 s Ž Vg q hnt . sin Ž V z . q Ž tg y hnV . sinh Ž t z . q Ž dh y bm . cosh Ž t z . y cos Ž V z . M14 s Ž Vd y h mt . sin Ž V z . q Ž td q h mV . sinh Ž t z . y Ž bn q gh . cosh Ž t z . y cos Ž V z . M21 s Ž Vb y hht . sin Ž V z . q Ž tb q hhV . sinh Ž t z . q Ž ygm y dn . cosh Ž t z . y cos Ž V z .

Ž A1.

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M22 s V 2 q b 2 y m2 y n 2 cosh Ž t z . q t 2 y b 2 q m2 q n 2 cos Ž V z . M23 s y Ž htd q h mV . sin Ž V z . q Ž h Vd y mt . sinh Ž t z . q Ž bg y nh . cosh Ž t z . y cos Ž V z . M24 s Ž htg y nV . sin Ž V z . y Ž h Vg q nt . sinh Ž t z . q Ž bd q hm . cosh Ž t z . y cos Ž V z . M31 s Ž Vg q hnt . sin Ž V z . q Ž tg y hnV . sinh Ž t z . q Ž bm y dh . cosh Ž t z . y cos Ž V z . M32 s Ž htd q mV . sin Ž V z . y Ž h Vd y mt . sinh Ž t z . q Ž bg y nh . cosh Ž t z . y cos Ž V z . M33 s V 2 q g 2 y m2 y h 2 cosh Ž t z . q t 2 y g 2 q m2 q h 2 cos Ž V z . M34 s y Ž htb q hV . sin Ž V z . q Ž h Vb y ht . sinh Ž t z . q Ž gd y nm . cosh Ž t z . y cos Ž V z . M41 s Ž Vd y h mt . sin Ž V z . q Ž td q h mV . sinh Ž t z . q Ž bn q gh . cosh Ž t z . y cos Ž V z . M42 s y Ž htg y nV . sin Ž V z . q Ž h Vg q nt . sinh Ž t z . q Ž bd q hm . cosh Ž t z . y cos Ž V z . M43 s Ž htb q hV . sin Ž V z . y Ž h Vb y ht . sinh Ž t z . q Ž gd y nm . cosh Ž t z . y cos Ž V z . M44 s V 2 q d 2 y n 2 y h 2 cosh Ž t z . q t 2 y d 2 q n 2 q h 2 cos Ž V z .

Ž A2.

where h is the signŽ. function of the dot product x P y defined by:

°1 ¢y10

h s sign Ž x P y . s~

xPy)0 xPys0 xPy-0

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x

C.S. Brown, A.E. Bak, Opt. Eng. 34 Ž1995. 1625. J.L. Wagener, D.G. Falquier, M.J.F. Digonnet, H.J. Shaw, J. Lightwave Technol. 16 Ž1998. 200. J.F. de Boer, T.E. Milner, J.S. Nelson, Opt. Lett. 24 Ž1999. 300. M.W. Shute Sr., C.S. Brown, J. Jarzynski, J. Opt. Soc. Am. A 14 Ž1997. 3251. H.Y. Kim, E.H. Lee, B.Y. Kim, Appl. Opt. 36 Ž1997. 6764. R.C. Jones, J. Opt. Soc. Am. 31 Ž1941. 488. R.M.A. Azzam, J. Opt. Soc. Am. 68 Ž1978. 1756. R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland, New York, 1977, pp. 1–65, 490–492. M. Born, E. Wolf, Principles of Optics, 6th edn., Pergamon Press, New York, 1965, pp. 23–32, 544–555, 665–716. K. Ogata, Modern Control Engineering, 2nd edn., Prentice Hall, Engledwood Cliffs, 1990, pp. 724–1005.

Ž A3.