CAPITULO II: MARCO TEÓRICO
2.2. BASE TEÓRICA
2.2.1. AUTOESTIMA
Anon-deterministic scatterer is, in a broad sense, an optical system which cannot be de- scribed by a Mueller-Jones matrix. In this class fall any depolarizing optical systems, such as multi-mode optical fibers, particles in suspension, etc. It has been shown [61, 62] that it is possible to describe a non-deterministic optical system as an ensemble of deterministic sys- tems, in such a way that each realizationE in the ensemble is characterized by a well-defined Jones matrixT(E)occurring with probabilitypE ≥0. This ensemble representation, in turn, reflects some uncertainty in the description of the medium which can be explained in terms of unobserved degrees of freedom of the probe beam. Then, the Mueller matrixM of the system can be written as
M=Λ†(T⊗T∗)Λ, (4.14)
where the bar symbol denotes the average with respect to the ensemble representing the medium:
T⊗T∗=
∑
E
pET(E)⊗T∗(E). (4.15) At this point it is useful to introduce the auxiliary 4×4 Hermitian matrixHdefined as the system dynamical matrix [47]:
H=1 2 0,3
∑
μ,ν Mμν(σμ⊗σν∗), (4.16)4.2 Mueller-Stokes formalism where TrH=1, Mμν are the elements of the Mueller matrix andσi(i=0,1,2,3) the nor-
malized Pauli matrices (see Eq. 4.8). His by definition, positive semi-definite, that is, all its eigenvalues{λ0,λ1,λ2,λ3}are non-negative. It is possible to demonstrate [46] that any
Mueller matrix can be decomposed as a sum of at most four Mueller-Jones matrices:
M=
3
∑
i=0
λiMiJ, (4.17)
where λi are the eigenvalues of H (Eq. 4.16). This positive decomposition allows us to
give a probabilistic interpretation to λi by considering each of them as the probability of
occurrence of the Mueller-Jones matrixMiJ. Further, it is through (Eq. 4.17) that the statistical nature of a depolarizing process is revealed, and it becomes clear that two different media can be described by two different ensembles of Mueller-Jones matrices with exactly the same coefficientsλi. These eigenvalues can, in turn, be combined to form a single scalar quantity
that is a measure of the polarimetric disorder added to the field by the system, i.e., the entropy of the mediumEM: EM=− 3
∑
ν=0 λνlog4(λν). (4.18)Moreover, it is possible to show that the index of depolarization (or depolarizing power)
DMof the medium can be written as: DM= 1 3 4 3
∑
ν=0 λ2 ν−1 1/2 . (4.19)For a non-depolarizing system,EM=0,DM =1, andH has a single eigenvalue equal
to one, and the rest equal to zero. This means that its corresponding Mueller matrix is a Mueller-Jones matrix, as expected. Conversely, for a totally depolarizing mediumEM=1, DM=0 andHhas four eigenvalues equal to 1/4.
In Ref. [50], a universal relation betweenEMandDMwas found. This universal character
can be explained by noticing that bothEMandDMonly depend on the four eigenvalues (λi) of H. By using the normalization condition Tr{H}=1, we see there are only three independent parameters. We choose then to writeEM(DM,λ1,λ2), where 0≤λ1,2≤1 are two independent
eigenvalues of H. By varyingλ1,2it is possible to obtain a whole domain en theEM−DM
plane (see Fig. 4.2). The analytical curves that determine this domain are detailed in Ref. [50]. Cuspid points in Fig. 4.2 separate different types polarization dependent scattering processes. In subsection 4.2.1, Eq. (4.6), we pointed out the analogy between the definition ofpolar- ization entropyand the von Neumann entropy of a quantum system. In the case of bipartite systems, the von Neumann entropy quantifies the degree of entanglement of the subsystems for pure states, and it is a measure of the degree of mixture for non-pure states. It is worth noticing that it is possible as well to relate thedegree of polarizationof a system of dimension
N, with its linear entropy, which is an alternative measure of the degree of mixedness of a system (and is generally easier to calculate than the Von Neumann entropy) [55,56]. Namely, given the density matrix of the systemρ, one can define the degree of polarizationPN by:
PN=
NTr{ρ2} −1
E
A
B
C
D
D
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 4.2: Numerically determined domain for the entropy (EM) vs index of depo-
larization (DM) corresponding to any scattering process. The solid lines are given by
analytical curves [50]. The four cuspid points are given byA= (0,1),B= (1/3,log43),
C= (1/√3,1/2), andD= (1,0).
Recalling that the linear entropy is defined as [55]:
SN= N
N−1[1−Tr{ρ
2}], (4.21)
and combining Eq. (4.20), and Eq. (4.21) it is simple to show that:
SN=1−PN2. (4.22)
The mathematical space where medium parametersEMandDMare defined is equivalent
to that of a quantum system with a Hilbert space of dimensionN=4, i.e., the polarization state space of two photons. Within this analogy,EMcan be interpreted as the von Neumann
entropy of a four dimensional (N=4) quantum system. Additionally, forN=4 we have DM=P4, andS4=1−D2M. Equations (4.21) and (4.22) are also valid in the case of a single
photon takingN=2.
In the next section we formally introduce the concept ofeffectiveMueller matrix which serves as a theoretical tool to describe our depolarization experiments.
4.3 The effective Mueller matrix