Nondepolarizing systems and degree of polarization

Volume 77, number 5,6

15 July 1990

OPTICS COMMUNICATIONS

NONDEPOLARIZING

AND DEGREE

POLARIZATION

Institute of Mathematical Sciences, Madras 600 113, India

Received 20 November 1989; revised manuscript received 7 March 1990

In polarization optics deterministic and nondeterministic systems are called nondepolarizing and depolarizing systems respectively. This unfortunate terminology has led to the belief that nondepolarizing systems do not reduce the degree of polarization and depolarizing systems do not increase it for any input state of polarization. We show that this belief is unfounded: nondepolarizing systems can indeed depolarize and depolarizing systems can polarize.

1. Introduction While the action of a linear deterministic system in polarization optics can be described by a complex 2 x 2 matrix called the Jones matrix, that of a nondeterministic system cannot be described by a Jones matrix but needs the (4 x 4 real) Mueller matrix. The latter can, of course, describe deterministic systems as well [ l-6 1. The question of a system being deterministic or nondeterministic is independent of the state of polarization of the light beam being processed by the system. On the other hand the degree of polarization 9, is a useful concept which serves to quantify the extent to which a light beam is polarized, i.e., the extent to which it is, roughly speaking, in a pure state of polarization so that 1 - 9 is a measure of the randomness as far as the polarization aspect of the beam is concerned. It may be tempting to believe that a deterministic system would always increase the degree of polarization while a nondeterministic system would necessarily decrease it #I. This belief is, if anything, only strengthened by the unfortunate terminology prevailing in the literature whereby nondeterministic systems are called depolarizing systems and deterministic systems are called nondepolarizing systems.

In this paper we show that this expectation is not consistent with the definition of depolarizing and nondepolarizing systems. In other words we show that nondepolarizing systems can decrease the degree of polarization and that depolarizing systems can increase the degree of polarization. It turns out that the family of nondepolarizing systems which decrease the degree of polarization at least for some input states is (infinitely) larger than the family of nondepolarizing systems which do not decrease the degree of polarization for any input state of polarization. In the light of these results we make a plea that the less misleading nomenclature of deterministic and nondeterministic systems be used in place of nondepolarizing and depolarizing systems respectively.

2. Polarization polarization

Let us consider a Cartesian coordinate system (x,, x2, x3) and let the light beam propagate along the positive x3 direction. Then the (pure) polarization state of a fully polarized light can be described by the Maxwell vector (also called Jones vector

,

‘I Such a belief is expressed, for example, on p. 14 I of the book by Azzam and Bashara [ 7 ] which is one of the best available expositions

of polarization

0030-4018/90/$03.50

optics.

0 Elsevier Science Publishers

states, optical systems and degree of

where E,, E2 are the components B.V. (North-Holland)

(2.1) of the transverse 349

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the Stokes vector. Using (2.6), we can readily invert (2.7 ) to obtain

electric field vector. On the other hand, the coherency matrix

as

@= (HZ+)

S,=tr(@r,),

(2.2)

can describe both partially coherent (mixed) and fully coherent (pure) states. Clearly, @ is related to E in exactly the same way in which the density matrix is related to the state vector in quantum mechanics. By construction the 2 x 2 complex matrix @ is hermitian and positive semidefinite, @+=@,

@>O.

E+E’=JE.

(2.4)

When the Maxwell vector E undergoes the transformation (2.4), the coherency matrix undergoes the transformation = J@J+ ,

(2.5)

which is bilinear in the elements of J. According to ref. [ 7 1, p. 142, “this equation represents the basic law of propagation of partially polarized light through linear nondepolarising optical systems”. Let Q be the 2X i! identity matrix and (T,, TV, TV) = (a3, CT,,aZ), where the a’s are the hermitian Pauli matrices in the usual representation. The r-matrices thus defined have several interesting properties, the most useful one being tr(r,7p)=2&ap,

a,p=O,

that is the r-matrices are a factor of fi. Since the set of hermitian matrices ear combination of these guaranteed to be real, @=$

i

saTa_;

Cl=0

The coefficients 350

(2.8)

showing that the coherency matrices and the Stokes vectors are in one-to-one correspondence and have identical information. In particular So = tr @ gives the intensity. While hermiticity of @ ensures, and is equivalent to, reality of S, the positive semidefiniteness of @ (namely tr @> 0, det @>O) is equivalent to

(2.3)

Singular coherency matrices, i.e. @ with det Q=O, correspond to pure states and positive definite coherence matrices correspond to mixed states. For unpolarized light @ is a multiple of the identity, and in all cases tr @ gives the intensity. A linear deterministic system (nondepolarizing system) is described by a 2x2 complex matrix J called the Jones matrix,

Q-Q’

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1,2,3,

(2.6)

trace orthogonal except for r-matrices form a complete we can expand 0 as a linand the coefficients are then

sO+si s?.-is3 S2 +iS, S, arranged

So-S,

>

as a column

(2.7) S is known

So>0

> s:,-s:-s:-s:~o.

(2.9)

Thus, Stokes vectors correspond to vectors within and on the positive light cone in a four-dimensional Minkowski space. The vectors on the surface of the light cone corresponding to saturation of the inequality in (2.9) represent pure states and those within represent mixed states. For unpolarized light S, = 0, o = 1, 2, 3 and such Stokes vectors are along the “time” axis (the So axis). Under the action of a linear system, the Stokes vector transforms as S+S’ =MS,

(2.10)

where M is a 4 x 4 real matrix called the Mueller matrix. Given a Jones matrix J transforming @ as in (2.5 ), the corresponding Mueller matrix can easily be derived. Such Mueller matrices derived from Jones matrices form a seven-parameter family #2 and correspond, of course, to nondepolarizing systems. The Mueller matrices on the other hand form a sixteenparameter family #3. Those Mueller matrices which lie outside the seven-parameter family cannot be derived from a Jones matrix and they correspond to depolarizing (nondeterministic) systems. One subtle aspect which hitherto seems to have been missed should be noted at this stage. One should not imagine that the coherency matrix formalism is inherently incapable of handling depolarizing systems. The transformation (2.10) we have written for The 2x2 Jones matrix has eight parameters (four complex parameters), but the overall phase of J does not enter the bilinear law (2.5) nor the Mueller matrix derived therefrom. M has to ensure that a Stokes vector obeying the conditions (2.9) is mapped into one obeying these conditions. But this requirement leads only to inequalities among the elements of M which do not reduce the number of parameters from sixteen.

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the Stokes vector is indeed the most general linear transformation. But that for the coherency matrix namely (2.5 ) is not the general linear transformation on @. The most general linear transformation for @ is @+ 0’ : q

= Q,,,JDk[ )

(2.11)

where Q is a “supermatrix” with the indices going over the values 1, 2. Our restricted law (2.5) corresponds to fil,,,, = J,,J:

(2.12)

.

By choosing (2.5) rather than (2.11) or, equivalently, by restricting Sz to the special form (2.12) we have a priori decided to handle only the nondepolarizing systems within the coherence matrix formalism even though it is possible to handle depolarizing systems as well. Incoherent superposition of light beams correspond to linear combination of coherency matrices (Stokes vectors) with real nonnegative coefficients, and vice versa. The concept of degree of polarization is best appreciated by recalling that any given partially polarized beam can be considered as a unique incoherent superposition of an unpolarized light and a fully polarized light,

By requiring the second hand side to correspond value of B to be

Stokes vector on the right to a pure state we fix the

9=[(S:+S:+S:)/S~]“*,

(2.14)

and thus the decomposition is made unique. The first vector is clearly unpolarized light. 9 is the degree of polarization and it is the ratio of the intensity of the fully polarized component to the total intensity. It is easy and useful to rewrite the expression (2.14) for 9 in terms of tr @ and det 0, the unitary invariants of the coherency matrix. To this end note that s’=1-(s~-s:-s:-s:)/s~. Since So=tr@ and S;-S:-S:-S:=4det@, immediately arrive at

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(2.15) we

P=[l-4(det@)/(tr@)2]1/2,

(2.16)

a result first derived by Wolf through a different proach [ 8 1.

3. Classification

ap-

scheme for Jones matrices

We wish to see how the degree of polarization is transformed by a nondepolarizing (i.e. deterministic) system. To this end it is advantageous to classify the Jones matrices in the following manner, keeping in mind that the overall phase of J is irrelevant for the present problem. (i) Singular Jones matrices. These are Jones matrices with vanishing determinant and they consitute a five-parameter family. Indeed, these Jones matrices can be parametrized as J=e’Y

(3.1)

We obtain three independent parameters from the real a, b, c, d subject to the constraint ad= bc and, suppressing the overall phase y, the phases (Y,p contribute two independent parameters to make the singular Jones matrices a five-parameter family. Let us call this family Cs. (ii) Nonsingular Jones matrices. These are Jones matrices with nonvanishing determinant and hence can be written in the form

J=A e’“V,

(3.2)

where V is a general complex 2 x 2 matrix with unit determinant. The V matrices are elements of the group SL( 2,C) and hence constitute a six-parameter family. Suppressing the overall phase cy the real positive ,l gives the seventh parameter to make the nonsingular Jones matrices a seven-parameter family. Let us denote this family CNS. (iii) Jones matrices which are multiples of unitary matrices. These are Jones matrices of the form

J=Ae’“U,

(3.3)

where U is a unitary matrix with unit determinant, and hence is an element of the three-parameter group SU( 2). Again suppressing the over all phase a! the Jones matrices in (3.3) constitute a four-parameter family with the real positive scalar 1 giving the fourth 351

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This four-parameter

family will be called

-+I.

Since SU(2) is a proper subgroup of SL(2,C) it is evident that Cu is a proper subset of CNS. Further, Cs and CNS are nonintersecting sets and so also are C, and Cu. Finally, the totality of Jones matrices c is the union of CNS and Cs. Thus, C=&u&s,

C” c%s,

C,nZ’NS=CSnCu=O,

(3.4)

where 0 is the empty set.

4. Nondepolarizing polarization

[ 1-4(det

,/p’=l,

for every 9 .

@‘)/(tr

@‘)2]‘/2.

(4.1)

Thus, every system belonging to family Cs is a polarizing system giving fully polarized output for every input state of polarization. Family Cu. In this case the Jones matrix is essentially a unitary matrix and is given by ( 3.3 ). In view of the invariance of trace and determinant under unitary transformation we have det @’ =A4 det 0.

$7 1= p

(4.6)

for every input state of polarization. Family ENS. The family Cu is a subset of C,, and we have already analysed the former. Thus it is sufficient to analyse CNs - Cu (i.e. elements of CNS lying outside C,). Such Jones matrices can be parametrized as [see eq. (3.2) ] J=AU2SU,,

(4.7)

S=

with p> 1. It is useful to parametrize erency matrix @ as

The relationship between @ and the hermitian positive semidefinite 6 is one-to-one and invertible. With the parametrizations (4.7) and (4.8) it is immediately seen that tr @‘=A2(p2a+pL2b) det @’ =A2 det @ .

P2=cT,P,c,=

(

;

‘:

>

.

(4.3)

If a light beam with coherence matrix belonging to this family is passed through the system, the output intensity tr(J@Jt) is clearly zero and hence the *’ For a given J the unitary matrices the one-parameter mutes with PI.

the input coh-

(4.8)

where 1 is real positive scalar and P, is a projection matrix (Pf = PI ) #4. Every such Jones matrix annihilates a one-parameter family of coherency matrices parametrized by a real positive scalar /? and given by @=jxJtP2U,)

(4.5)

in (4.1) we find that

U, , (4.2)

352

(4.4)

Using these expressions

Family Cs. Since a Jones matrix belonging to this family has vanishing determinant, it can be written as J=A&P,

question of the value of 9’ does not arise. For any other input state of polarization tr CD’>O, but det @’ =A2 det @(det PI )2=0 since P, is singular. Hence from (4.1),

tr @’ =A2 tr @ ,

systems and degree of

With the aid of the classification in section 3, the transformation of the degree of polarization under the action of a nondepolarizing (i.e. deterministic) system can be easily analysed. It is convenient to present the analysis for each family separately, keeping in mind that in every case the output coherency matrix @’ is given by (2.5) and the degree of polarization .Y ’ of the output light by Y’=

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subgroup

U,, Ci, are not unique since of SU (2) generated by 0) com-

, (4.9)

Now, the expression p2a + ,u -2b $ a + b according as b/a 2 ,u2. Since a + b is indeed tr @, we see on substitution of (4.9) in (4.1) that the degree of polarization of the output light is greater than, equal to or less than that of the input according as b is less than, equal to or greater than ap’, (4.10)

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Thus, under the action of a Jones matrix JE (&s-Co) the four-parameter family of input coherency matrices (labelled by the two real parameters a, b and the complex parameter c as in (4.8 ) ) divides into three disjoint sets specified, respectively, by b/a2 p*. The degree of polarization increases for the first, remains unaffected for the second and decreases for the third. The important point being made is that the subset of input states for which 9 is decreased by the system is a four-parameter family in just the same way as the subset for which 9 is enhanced. And this is true for every Jc (&S-C”). As a concrete physical situation #’ corresponding to the above results consider a light beam which is an incoherent superposition of x, and x2 linear polarization states with intensities a and b respectively so that the coherency matrix is

It is clear that 9’ can be less than, or equal to, or greater than 9 depending on the relative value of a/ b to I a I */ I /?I *. In particular the output light will be unpolarized (i.e. P’=O) if b/a= [al*/ I PI*. This simple physical situation illustrates our principal result in this section: a nondepolarizing system cannot only reduce the degree of polarization but also can convert partially polarized beams into unpolarized ones.

5. Depolarizing

A depolarizing (i.e. nondeterministic) system is one whose Mueller matrix cannot be derived from a Jones matrix. The most obvious example, of course, is the ideal depolarizer whose Mueller matrix is /l

(4.11) Assume that this beam is passed through a nondepolarizing (i.e. deterministic) system which attenuates x, and x2 components by different factors but does not mix them so that it has a Jones matrix of the form (4.12) with 1a 1# 1PI. An elementary example of such a system is Fresnel reflection (or transmission) at a (multilayer) dielectric film chosen with its s and p components to be the x, and x2 components respectively. The degree of polarization of the input beam is, from (2.16), P=la-bl/(a+b). The output beam is an incoherent superposition of x1 and x2 linear polarization states with intensities jcrl*a and I /?I *b respectively so that its coherency matrix is

O ).

(4.13)

IPl*b

The degree of polarization from (4.1), P’=II(YI*aIf5 The author construction.

of the outcoming

Ij?12bll(la12a+ is grateful

light is,

Ijll’b) .

to the referee

for suggesting

this

systems and degree of polarization

0

0

o\

I I

M=OOOO D 0

0

0

0’

0

0

0

0

(5.1)

To obtain another example follow MD by a right circular polarizer MR and consider the combined system whose Mueller matrix is

MCMRMDZf

/l

0

0

o\

;

;

;

;

i 1

0

0

0i

.

(5.2)

It is a depolarizing system as well. In section 4 we have seen that nondepolarizing systems can decrease the degree of polarization. It turns out that depolarizing systems can increase the degree of polarization. A striking example is the depolarizing system whose Mueller matrix M is given in (5.2). Whatever be the input, this system produces a right circularly polarized output and the degree of polarization of the output light 9’ = 1 irrespective of the value of the input degree of polarization 9. It should not be thought that the above behaviour is just an analytic result valid only for systems involving ideal depolarizer MD. The phenomenon of a depolarizing system increasing the degree of polarization can very well occur in practically realizable systems. As an example consider the following 353

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experimentally measured Mueller matrix sponding to the Howell system [ 91,

0.7599 -0.0573 = i

-0.0623 0.4687

0.0384

-0.1714

0.1240

-0.2168

0.0295 -0.1811 0.5394 -0.0120

COMMUNICATIONS

Ms corre-

0.1185 -0.1863 0.0282

’

0.6608 1 (5.3)

It has been shown to correspond to a depolarizing system [ 1,2 1. However, this system can indeed increase the degree of polarization. To see this consider an unpolarized light of unit intensity being passed through the system. The Stokes vector of this beam has S, = 1, S, = S, = S, = 0 so that the degree of polarization .Y = 0. The Stokes vector of the outcoming light is

S’ =

15 July 1990

in a well-defined sense that nondepolarizing systems reduce the degree of polarization at least for some input states, with probability one. We have also shown that each system belonging to Z&s-& reduces the degree of polarization for a characteristic four-parameter subset of input states out of the fourparameter family of all possible input states. We have already noted that the phrases “nondepolarizing systems” and “depolarizing systems” are misleading and that this has led to erroneous claims. Moreover, it is fairly sensible to call the phenomenon of reduction in the degree of polarization “depolarization” and that of enhancement “polarization”. In view of this and in view of our results above, we conclude with the plea that the phrases deterministic and non-deterministic systems be used in place of the misleading phrases nondepolarizing and depolarizing systems respectively. Or else, we have to face statements like: a nondepolarizing system can depolarize and a depolarizing system can polarize!

(5.4) Acknowledgement

We see from (2.14) that its degree of polarization 9 = 0.187. Thus, the depolarizing systems Ms (partially) polarizes an input unpolarized light.

The author thanks Professor Emil Wolf for illuminating discussions. He is grateful to the referee for insightful comments leading to the demonstrations of the results presented at the end of sections 4 and 5.

6. Concluding remarks We have shown that nondepolarizing systems can reduce the degree of polarization and that depolarizing systems can increase the degree of polarization. The family of nondepolarizing systems which do not reduce the degree of polarization is shown to be the five-parameter family CsuC,. In contrast the family of nondepolarizing systems which reduce the degree of polarization is the seven-parameter family CNS-CU. That is, the fraction of nondepolarizing systems which do not reduce the degree of polarization for any input state is zero. Thus, we can state

354

References [ 11 R. Simon, Optics Comm. 42 ( 1982) 293. [2] R. Baralat, Optics Comm. 38 (1981) 159. [ 31 E.S. Fry and G.W. Kattawar, Appl. Optics 20 ( 198 1) 28 I 1. [4] J.J. Gil and E. Bernabeu, Optica Acta 32 (1985) 259. [ 51 K. Kim, L. Mandel and E. Wolf, J. Opt. Sot. Am. A4 (1987) 433. [6] R. Simon, J. Mod. 34 (1987) 569. [ 7 ] R.M.A. Azzam and N.B. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977 ). [8]E.Wolf,NuovoCim.13(1959)1165. [ 9 ] B.J. Howell, Appl. Optics 18 ( 1979) 809.