Polarization entropy transfer and relative polarization entropy

I February

1996

OPTICS

COMMUNICATIONS ELSEVIER

Optics Communications

123 ( 1996) 44348

Polarization entropy transfer and relative polarization entropy Richard Barakat Aiken Computation Laboratory. Harvard Universiry, Cambridge, MA 02138, USA Electra-Optical Research Center, Tufts University, Medford. MA 02155, USA Received 16 December

1994; revised version received 28 July 1995

Abstract The concept of polarization entropy is extended to include its transferby an optical system (or scattering medium) characterized by a Jones matrix (or equivalently, a Mueller-Jones matrix) and also by a general Mueller matrix. It is shown that physical realizability for a deterministic, passive medium governed by a Jones matrix implies enpolarization (i.e., output degree of polarization is greater than input degree of polarization) and that the output entropy is less than the input entropy. Thus enpolarization leads to a decrease in the uncertainty of the output relative to the input as characterized by the respective degree of polarization. General Mueller matrices can depolarize (i.e., output degree of polarization is less than input degree of polarization). leading to an output entropy that is greater than the input entropy. Thus depolarization leads to an increase in the uncertainty of the output relative to the input. An entropic measure of the “distance” between input and output degrees of polarization. the relative entropy, is introduced and studied.

1. Introduction

and preliminaries

The von Neumann entropy of a quantum system characterized by a density matrix p is defined as [ 1,2] H= - tr(plnp)

= -

C h,lnh,, . w

(1)

The A,, are the eigenvalues of p; they are real, nonnegative and sum to unity (because trp = 1) . The density matrix has a partial analog in the spectral density matrix a(w) occurring in the theory of partially polarized light [ 31. Both are hermitian and positive semidefinite. The density matrix satisfies a differential equation and thus possesses dynamics; the spectral density matrix, on the other hand, is a kinematical quantity that describes the second-order statistical properties of the interacting beams. It possesses no dynamics. In addition, the density matrix characterizes the states of the quantum systems, whereas the spectra1 density matrix only characterizes the second0030.4018/96/$12.00

0 1996 Elsevier Science B.V. All rights reserved

SSD~,10030-4018(95)00586-2

‘order statistics of the system (at frequency w). Consequently the spectral density description is only complete when the underlying stochastic structure of the electromagnetic field amplitudes is Gaussian [ 3 3 Fortunately, the Gaussian case is of most interest because it covers light of thermal origin. Given these caveats, it is still possible to define an entropy using the spectral density matrix provided that it is properly normalized. To this end define A(w)=

- l @(w) , tr@( 0)

(2)

thus trA( w) = 1 as required. We define the entropy of N pencils (i.e., thin beams) of light to be H(w) = -tr[A(w)InA(W)

J,

(3)

or

H= - 5 h,lnh,, ll=l

(4)

444

R. Barakat/Oprics

Communications 123 (1996) 443-448

where the A,, re the eigenvalues of A. The case of N beams has been studied at some length in Barakat and Brosseau [ 41 using n-fold polarization measures developed by Barakat [5]. Unfortunately they term the entropy as given in Eq. (3) a von Neumann entropy. It is an entropy, but not a von Neumann entropy! Rather H partakes some of the essential feature of the Shannon entropy [ 6,7 ] in that the eigenvalues of A, being nonnegative and summing to unity, possess the formal properties of probabilities. Thus H, at a fixed w, acts somewhat like a Shannon entropy, in spite of being derived from a normalized spectral density matrix. In fact, it appears to lie between a true von Neumann entropy and a true Shannon entropy. I have decided to term H(w),the polarization entropy. Thus the polarization entropy is a kind of informational entropy (in the spirit of Shannon) because it only measures the uncertainty in the context of second-order statistics (i.e., degree of polarization) rather than being a full description of the wave in the quantum mechanical sense. Of particular importance in the case N= 2 which can be interpreted as a plane wave situation. In this case A(w) =-

1 2&,(w)

&r(w) +S,(o)

&(o)

%(w)+&(w)

Se(o) -S,(o)

-i&(w) ’

(5) where the Si( w) are the spectral Stokes parameters

[ 31

.Si(~)=tr(@(w)uj),

(6)

and the oj are the Pauli matrices. These Stokes parameters should not be confused with the temporal Stokes parameters [ 81. The eigenvalues of A are hi.,=

itrA+

f[ (trA)2-4detA]1’2=

i( 1 &-P(w))

, (7)

where P(o)

is the spectral degree of polarization

P(o)= -J- [s:(o)+s~(w>+s~(w)]"2,(8) Ww) with OgP( w) G 1. When the eigenvalues tuted into Eq. (4)) one obtains H=-

[(y)ln(q)

are substi-

088 8

0

045

Fig. I Polarizationentropy as a function of thedegreeof polarization. The limiting

H(0)=ln2,

values are: H(l)=O,

(9)

(10)

with a monotone decay from completely unpolarized light to completely polarized light see Fig. 1. It is also possible to study a version of the polarization entropy in terms of the covariance matrix of the wavefield. For a plane wave, the covariance matrix is C-_ 1 (%(r))+(Si(r)) 2 (&(r))+iMt))

(s2(t))-i(&(r)) (so(r))-(S1(t))

.’ (11)

where (Sj(t)) are the mean values of the temporal Stokes parameters. The formal mathematical manipulations are the same as for the corresponding spectral density matrix. However, should we wish to study polarization entropy transfer, we cannot avail ourselves of Eq. ( 12)) but rather must consider a matrix convolution between input and output, see Brosseau and Barakat [ 91 for the details. The equivalent of Eq. (9)) but in the context of the covariance matrix (i.e., temporal Stokes parameter) is given in [ IO]. See Brosseau [ 111 or another, but more complicated, derivation of Eq. (9) again in the context of the covariance matrix. 2. Spectral density/Jones

+ (y)ln(y)].

180

matrix description

The problem we now address is the determination of the output polarization entropy, H(P,), given the input

R. Barakat/Oprics

Communications

polarization entropy, H( Pi), and the Jones matrix characterizing the medium at frequency w, J( w). The input and output spectral density matrices 4pi( w) and eo( w) are related by [ 31 @n(w) =J(~)~(o)J*(w)

?

(12)

123 (1996) 443448

44s

polarization. If H,, < Hi, as determined by a true density matrix, then we would have a Maxwell’s Demon [ 13 J operating, which we certainly do have!

3. Stokes vector/Mueller-Jones

matrix description

where (13)

An alternative, but equivalent, description of the spectral density matrix/ Jones matrix of the problem is the Stokes vector/Mueller-Jones matrix formalism S=M4,

(14)

(19)

where 9 and S are the input and output spectral Stokes vectors

Consequently, A,, = g ‘J&J*

,

where g is the gain (transmittance) g = SO/
(15) at frequency

(20)

w: (16)

Upon taking the determinant of both sides of Eq. ( 12) we have the polarization transfer equation [ 31

and M is the Mueller-Jones matrix (not to be confused with the general Mueller matrix to be discussed shortly)

(I-P:)=g.-‘(detJJ*)(l--P:).

M=A[J@j]A-'

(17)

A Jones matrix characterizing a passive, deterministic medium is said to be physically realizable at frequency w if a) 0 Pi( w) . This follows directly from the polarization transfer equation rewritten in the form

(18)

.

(21)

Here j is the Jones matrix with all the elements taken to be complex conjugate, but not transpose, and A is rl00 10 0

i

0

li -1

-i

0

.

(221

matrix satisfies [ 121

The Mueller-Jones M*GM=

A-r=‘A* ?

[det(M)]“*G,

(23)

where and noting that the right-hand side is always nonnegative. Thus a physically realizable deterministic medium cannot decrease the output degree of polarization relative to the input degree of polarization. In the important case where J is unitary, then P, = Pi. We will term the situation P,, > Pi enpolarization, and the situation P,, < Pi depolarization. Thus physical realizability for a system governed by a spectral density matrix/Jones matrix cannot depolarize. This consequence, along with Fig. 1, leads to Ho < Hi; all we have done is to decrease the uncertainty as measured by the degree of

rl

0

0

01 (24)

The polarization

transfer equation now reads

(l-P~)=g-2]detM]1’2(1-P~).

(25)

All remarks made for the spectra1 density matrix/ Jones matrix description hold mututis mutundis for the

446

R. Barakat/Optics

Communications 123 (19961443448

Table I Output degree of polarization and output polarization entropy enpolarization, the unstarred entries denote depolarization.

,*

,

0

0

0

2 3 4

I

I

0

0

I

I 0 0 0.5 0 0 0.707 0.4 0

0 0 0 0 0.5 0 0.0 0.4 0.4

0

6 5 7:”

8:s 9 IO 11”

I i I I I 1

Stokes vector/Mueller-Jones expected.

4. Stokes vector/general description

0.760 0.698 0.822 0.878 0.64 1 0.788 0.834 0.844 0.716 0.747 0.819

-0.057 0.411 - OS26 0.244 0.129 0.084 -0.241 -0.189 0.274 0.058 - 0.204

description,

as

I -1 0.5 0.5 0.707 0 0 0.4

matrix

for various input cases for the Howell matrix. The starred entries denote

Mueller matrix

Let us return to Eq. (19) and consider M to be a general Mueller matrix (i.e., it cannot be derived from a Jones matrix). Since a general Mueller matrix is obtained from experimental measurements [ 141, it always satisfies the condition: O
1

implies

O
I .

1 -Pi)

_

(27)

Consequently .S;( 1 -P;)

=4”(M*GM)4.

(28)

0.124 -0.093 0.341 0.785 - 0.537 0.346 0.448 0.59 1 - 0.029 0.033 0.384

0.693 0 0 0.417 0.417 0.485 0.523 0.523 0.629 0.629 0.629

0.520 0.582 0.274 0.692 0.615 0.430 0.615 0.483 0.591 0.509 0.592

0 1

I 1

I 0.707 0.707 0.707 0.707 OS66 0.566

0.187 0.634 0.804 0.939 0.861 0.454 0.723 0.739 0.402 0.264 0.622

As can be seen, the output degree of polarization now depends upon the details of the input through the Yj rather than only upon Yr, and Pi. Note that when M is Mueller-Jones, then Eq. (28) reduces to the polarization transfer equation upon substituting Eq. (23) into the right-hand side. The easiest way to demonstrate the complexity of the situation as regards P, and Ho is to consider two measured Mueller matrices taken from the literature. The first Mu is from Howell [ 1.51,which experiments concern passage of partially polarized light through a collimator-radiometer system:

(26)

The situation with respect to the general Mueller matrix is quite different from the two previous cases. A general Mueller matrix admits both enpolarization (P,, > Pi) and depolarization (P,
0.038 0.133 0.200 0.067 0.010 - 0.033 0.322 0.058 - 0.083 0.186 0.265

M”=

0.7599 - 0.0573 0.0384

- 0.0623 0.4687 -0.1714

0.1240 -0.2168

0.0295

0.1185

-0.1811

0.1863

0.5394 -0.0120

0.0282 0.6608

'

The second, MR, is from Ramsey [ 161, which experiments deal with scattering of partially polarized light from a randomly rough surface: 1.OOOO - 0.5261 lVZn=

-0.4131 - 0.3908

0.5508

0.1205 0.1924

0.1279 - 0.2762

0.5560

- 0.0417

0.2983

- 0.0498

0.0923

- 0.438 1 - 0.2684

’

Note that Mu and MR represent two very different scenarios. Tables 1 and 2 show the numerical results. The starred entities indicate enpolarization, while the unstarred entities indicate depolarization. For example, entries 6 and 7 in Table 1 possess the same input degree

R. Barakat / Optics Communications

123 (1996) 443-W

44-l

Table 2 Output degree of polarization and output polarization entropy for variousinput cases for the Ramsey matrix.The starredentriesdenote enpolarization.the unstnrredentriesdenotedepolarization. ’

’

IN

.I’,

./2

I

SC,

s,

&

-0.413

-0.391

, :,:

I

0

0

0

1.000

2

I

I

0

0

0.474

ST

6

H,>

p,

-0.050

p<,

0.693

0.520

0

0571

0.043

0

0.582

I

0.463

0.689

0.219

0

0.274

1

0.844

0.002

0.016

0.417

0.692

0.707

0.04s

0.417

0.615

0.707

0.389

0.485

0.430

0.622

0.691

0.138

0.165

0.137

3

I

0

0

-I

I

0.707

0

0

0.628

-0.024

5

I

0.5

0.5

0

0.677

-0.042

-0.134

6

I

0

0

0.920

-0.241

-0.576

7

I

0.4

0.4

0

0.741

-0.116

-0.185

-0.188

0.523

0.615

0.566

0.389

*':

I

0

0.4

0.4

1.000

-0.446

-0.288

-0.332

0.523

0.483

0.566

0.625

9."

I

0.25

0.2.5

0

0.838

-0.227

-0.262

-0.136

0.629

0.591

0.354

0.445

I0:"

I

0

0.25

0.25

1.000

-0.434

-0.327

-0.226

0.629

0.509

0.354

0.587

, , :‘:

I

0.25

0

0.25

0.901

-0.345

-0.177

-0.094

0.629

0.592

0.354

0.443

-I

-0.622

0.872

of polarization, yet entry 7 is enpolarizing while entry 6 is depolarizing. Thus the output is more uncertain ( ix., has greater entropy than input) for entry 6, yet entry 7 has less uncertainty than the input. These results indicate, not surprisingly, that information (as measured by the polarization entropy) is lost when depolarization occurs. Reference is made to Brosseau [ 17,181 for an alternative approach to entropy transfer.

5. Relative polarization

We now introduce a second entropy function, the relative entropy which measures the “distance” hetween two discrete probability distributions: (pO, p,, ...) and (y,,. ql. .. ) The relative entropy of Q with respect to P is defined to be [ 71 WQlP)

E c q,ln(q,lp,) ,I :. 0

.

(29)

It can be shown that D(QIP)

(30)

>O.

with equality if and only if Q = P. The relative entropy is not a true distance because D(Q(P)

+D(P(Q)

.

(31)

and D does not satisfy the triangle inequality required of a true distance. Nevertheless it is useful to regard D( Q 1P) as a measure of the “distance” between Q and P.

0.117

By analogy to Eq. (29), we define the relative polarization entropy of two normalized spectral density matrices A2 and A, characterizing two sets of N pencils of radiation to be D(&jA,)

=tr[A21nA2-A,lnA,)]

,

(32)

provided that A, possesses a finite inverse (in our problem A, cannot characterize a completely polarized situation). If A,(‘) and A:*’ are the respective eigenvalues of A, and A2, then D(AJA,)

entropy

-0.223

= 5 hj2’ln(hj2’lhj”) j=l

.

(33)

In the case of a plane wave (N = 2)) this reduces to D(A,lA,)=

+ (y)ln(i).

(y)ln(%)

(34)

The relative entropy is a monotone, increasing function of P, with its maximum occurring when A2 is completely polarized and A, is completely unpolarized. This is the type of answer we would expect. in that the maximum “distance” between A2 and A, occurs when one is completely polarized and the other completely unpolarized. If Ai = A, and A, = A,, then D( A,, ( Ai) is the relative polarization entropy of the input and output polarizations of a plane wave that has traversed a scattering medium or non-image forming optical system. If the scattering medium is governed by a unitary Jones matrix, then

448

D(A”IA,)

R. BarakatlOptics

=o.

Communications

123 (19961443--148

[61 R.

(35)

Any nonunitary Jones matrix, such as for an anisotropic absorber, will cause the relative polarization entropy to be positive. In like fashion, any nonunitary general Mueller matrix (whether enpolarizing or depolarizing) will cause the relative entropy to be positive with the magnitude depending on M and 9.

References ] I ] .I. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, NJ, 1953). 121D. ter Haar, Statistical Mechanics (Reinhatt, New York, 1956)) Chapter 7. [ 3 1R. Barakat, J. Opt. Sot. Am. 54 (1964) 920. 141 R. Barakat and C. Brosseau, J. Opt. Sot. Am. A 10 (1993) 529. ]5]R.Barakat,Opt.Acta30(1983) 1171.

Gallagher, Information Theory and Reliable Communication (Wiley, New York, 1968). [7] T. Cover and J. Thomas, Elements of Information Theory (Wiley, New York, 1991). [S] M. Born and E. Wolf, Principles of Optics, 6th edition (Pergamon, Oxford, 1983) [9] C. Brosseau and R. Barakat, Optics Comm. 84 ( 1991) 127. [lo] EL. O’Neill, Introduction to Statistical Optics (AddisonWesley, Reading, MA, 1963). [ 111 C. Brosseau,C.R. Acad. Sci. Paris B310 (1990) 181. [ 121 D.G. Anderson and R. Barakat, J. Opt. Sot. Am. 11A (1944) 2305. [ 131 H. Leff and A. Rex (eds.), Maxwell’s Demon: Entropy, Information, Computing (Princeton Univ. Press, Princeton, NJ, 1990). [ 141 J. Gerrard and M. Burch, Introduction to Matrices in Optics (Wiley, New York, 1975). Chapter 6. [ 151 B. Howell, Appl. Optics 18 (1979) 1809. [ 161 D. Ramsey, Thin film measurements on rough surfaces using Mueller matrix ellipsometry, Ph.D. dissettation (University of Michigan, Ann Arbor, MI, 1985). [ 171 C. Brosseau, Optik 85 (1990) 180. [ 181 C. Brosseau, Optik 88 (1991) 109.