Download Characteristic properties of Mueller matrices. JJ Gil

Journal Optical Society of America A, 17, 328-334, 2000
Characteristic properties of Mueller matrices
José J. Gil
I.C.E. Universidad de Zaragoza, 50009 Zaragoza, Spain
Abstract:
A complete and minimum set of necessary and sufficient conditions for a real 4x4 matrix to
be a physical Mueller matrix is obtained. An additional condition is presented to complete
the set of known conditions, namely the four conditions obtained from the non-negativity
of the eigenvalues of the Hermitian matrix H associated with a Mueller matrix M and the
Transmittance Condition.
2
as being
Using the properties of H, a demonstration is also presented of Tr ( M T M ) = 4m00
a necessary and sufficient condition for a physical Mueller matrix to be a pure Mueller
matrix.
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Journal Optical Society of America A, 17, 328-334, 2000
1. Introduction
A high degree of knowledge of the relationships between the elements of a Mueller matrix
is important for analyzing the information contained in such matrices obtained from
experimental measurements. The contributions of various authors who have dealt with this
subject provide several sets of characteristic conditions. Thus, one criterion is the nonnegativity of the four eigenvalues of the Hermitian matrix H associated with a Mueller
matrix M1. Another is that the Mueller matrix of any passive optical system must satisfy
the “Transmittance Condition”, i. e. g f ≤ 1 , where g f is the maximum transmittance with
regard to all possible incident polarization states2. Other conditions have been derived from
the physical premise of the degree of polarization of outgoing light being less than or equal
to unity (i. e. the Mueller matrix not “overpolarizing”)3. These and other relevant results
have been obtained from the properties of the spectrum and eigenvectors of GMTGM,
where G is the diagonal matrix D(1,-1,-1,-1)3-6.
In this paper we analyze the general properties of "physical Mueller matrices" defined as
ensemble averages7 of passive linear elements and we find that the transmittance condition
for all these elements results in two inequalities: the well-known Transmittance Condition
for the whole system as well as a new necessary condition. The mathematical expression of
this new condition is symmetrical to the Transmittance Condition in the sense that it can be
written by interchanging the indices of the Mueller matrix elements involved.
Furthermore, the characteristic properties of H allow us to obtain a demonstration of the
2
“Purity Criterion” based on the equality8-10 Tr ( M T M ) = 4m 00
.
2. Transformation of polarization by linear passive pure systems
Let us consider the effects of a deterministic-nondepolarizing optical system on the
polarization properties of an incident light beam. We use the term “deterministicnondepolarizing” or “pure” in the sense of totally polarized incident light always emerging
totally polarized i. e. the system can be represented by a Jones matrix. It should be noted
that, as Simon has pointed out11, the degree of polarization of a partially polarized light
beam could decrease when it interacts with some kinds of deterministic optical systems (for
example diattenuators). Furthermore, in some circumstances, deterministic optical systems
can depolarize totally polarized polychromatic light12 and the system can not be represented
by a Jones matrix.
Some previous papers13-15,9 include the main formulation presented in this section, but it is
worth reproducing it here in order to introduce the notations and for clarity of subsequent
calculations.
Let us consider the analytical signal representations ε 1 (t ), ε 2 (t ) of the two mutually
orthogonal components of the electric field in a plane perpendicular to the direction of
propagation, so that
~
ε 1 (t ) = E1 (t ) + iE1 (t ) ,
~
ε 2 (t ) = E 2 (t ) + iE 2 (t ) ,
(1)
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Journal Optical Society of America A, 17, 328-334, 2000
~
~
where E1 (t ) , E 2 (t ) , are the real components of the electric field, and E1 (t ) , E 2 (t ) , are its
Hilbert transforms.16,17
When light incides on a linear passive optical system, the transformation of the field state
can be written as
ε ′(t ) = Tε(t ) ,
(2)
where ε', ε, are the emerging and incident Jones vectors, and T is the Jones matrix of the
system.
In general both emerging and incident fields can fluctuate, and the polarization state can be
described by the polarization matrix (or coherency matrix)16,18, i. e., the 2x2-covariance
matrix of the two variables ε 1 (t ), ε 2 (t ) ,
⎡ ε1ε1*
Φ = ε ⊗ ε+ = ⎢
*
⎢⎣ ε 2 ε1
ε1ε *2 ⎤
⎥,
ε 2 ε *2 ⎥⎦
(3)
where the angular brackets denote time averaging, ⊗ denotes the Kronecker product, ε+
denotes the complex conjugate transpose of ε, and ε* stands for the complex conjugate of ε.
Φ and the corresponding Stokes parameters are related by19,16
Φ=
1 3
1⎡s +s
sk σ k = ⎢ 0 1
∑
2 k =0
2 ⎣ s 2 + is3
s 2 − is3 ⎤
,
s0 − s1 ⎥⎦
(4)
where σ0 is the identity matrix and σi (i = 1, 2, 3) are the Pauli matrices
⎡1 0⎤
⎡1 0 ⎤
⎡0 1 ⎤
⎡0 − i ⎤
, σ1 = ⎢
, σ2 = ⎢
, σ3 = ⎢
σ0 = ⎢
⎥
⎥
⎥
⎥.
⎣0 1 ⎦
⎣0 − 1⎦
⎣1 0⎦
⎣i 0 ⎦
(5)
The polarization matrix of the emerging beam is
+
Φ′ = ε ′ ⊗ ε ′ + = Tε ⊗ (Tε ) = Tε ⊗ ε +T + = T ε ⊗ ε + T + = T ΦT + .
(6)
In order to make subsequent straightforward calculations, we will use the following
alternative notation for the elements of Φ9 (which we will also use, where convenient, for
other 2x2 matrices)
ϕ 0 ≡ ϕ11 , ϕ1 ≡ ϕ12 , ϕ 2 ≡ ϕ 21 , ϕ 3 ≡ ϕ 22 .
(7)
The elements of the polarization matrix can also be written as a column-vector ϕ defined by
ϕ = ε ⊗ ε ∗ with elements ϕi (i = 0, 1, 2, 3). This polarization vector ϕ and the
corresponding Stokes vector s are related by
s = Aϕ ,
(8)
where
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Journal Optical Society of America A, 17, 328-334, 2000
⎡1 0 0 1 ⎤
⎢1 0 0 − 1⎥
⎥.
A=⎢
⎢0 1 1 0 ⎥
⎥
⎢
⎣0 i − i 0 ⎦
(9)
The polarization vector ϕ′ of the emerging beam is given by
ϕ ′ = (T ⊗ T ∗ ) ϕ ,
(10)
where the pure system is represented by the “complex pure Mueller matrix” 13
V ≡ (T ⊗ T ∗ ) .
The corresponding pure Mueller matrix (also called Mueller-Jones matrix) N of the pure
optical system is related to V and T by13-15
(
)
N = AVA −1 = A T ⊗ T ∗ A −1 .
(11)
For reasons of clarity, we will use the following notation to distinguish between the
different types of matrices: L for complex Mueller matrices in general; M for Mueller
matrices in general and, in the case of pure systems, denoting the matrices by V and N
respectively.
As several authors have pointed out2,15,18,20-23 the condition for the elements of the Jones
matrix to represent a physically realizable optical system (assumed to be deterministicnondepolarizing) arises from the physical restriction that the intensity transmittance (gain)
g, defined as the ratio between the intensities of the emerging and incident light beams,
must be limited to the interval 0 ≤ g ≤ 1. This “Transmittance Condition” can be written in
function of the elements of T as follows
(
1 ⎧⎪
2
⎨ t11 + t12
2 ⎪⎩
2
2
+ t 21 + t 22
2
) + ⎡⎢⎣(t
2
11
− t12
2
2
+ t 21 − t 22
) + 4t
2 2
1
2
⎤ 2 ⎫⎪ ≤ 1 . (12)
∗
∗
t
+
t
t
⎬
11 12
21 22 ⎥
⎦ ⎪⎭
It should be noted that the condition 0 ≤ g is directly satisfied. The above condition is
therefore sufficient and no additional conditions are required for a 2x2 complex matrix to
be a Jones matrix2.
We will now denote the maximum transmittance as gf. The corresponding condition
expressed in function of the elements of N(T) is the following
2
2
2
g f = n 00 + (n 01
+ n 02
+ n 03
) ≤ 1.
(13)
This condition can also be expressed as
2
2
g f = n00 + (n102 + n20
+ n30
) ≤1
(14)
because the following equality24 is satisfied for any pure Mueller matrix N
2
2
2
2
2
(n 01
+ n 02
+ n03
) = (n102 + n 20
+ n30
).
(15)
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Journal Optical Society of America A, 17, 328-334, 2000
3. Transformation of polarization by non-pure linear passive systems
In general, an optical system can present spatial inhomogeneities over the area covered by
the incident light beam as well as dispersive effects, producing depolarization. The
emerging light beam (in any given direction) is consequently composed of several
incoherent contributions, and the optical system can not be represented by means of a Jones
matrix. However the system can be considered as a set of pure elements, each one with a
well-defined Jones matrix, in such a manner that the light beam is shared among these
different elements.
Let I(i) be the intensity of the portion of light that incides on the “i” element. The ratio
between I(i) and the intensity I of the whole beam is denoted by a respective coefficient pi
so that
I (i )
pi =
,
I
∑p
i
= 1.
(16)
i
Now we denote the Jones matrices, the complex Mueller matrices and the Mueller matrices
representing the “i” element by T(i), V(i) and N(i) respectively. Then, the Jones vector of the
wavelet emerging from the “i” element in the considered direction is
ε ′i (t ) = T (i ) ( pi ε (t )) ,
(17)
and the corresponding polarization vector is
ϕ′i ≡ ε ′i ⊗ ε ′i
∗
= (T (i ) pi ε ) ⊗ (T (i ) pi ε ) ∗
= (T (i ) ⊗ T (i )∗ ) pi (ε ⊗ ε ∗ ) = pi (T (i ) ⊗ T (i )∗ ) (ε ⊗ ε ∗ ) = piV (i )ϕ .
(18)
It should be noted that this formulation could be applied to different phenomena such as
refraction, reflection or scattering, providing that the elements of the target are passive.
The polarization state of the complete emerging beam is given by the incoherent
superposition of the beams emerging from each element. Thus, the polarization vector of
the emerging beam is
[
]
⎛
⎞
ϕ′ ≡ ∑ ϕ′i = ∑ pi (T (i ) ⊗ T (i )∗ ) (ε ⊗ ε ∗ ) =∑ piV (i )ϕ = ⎜ ∑ piV (i ) ⎟ϕ ,
i
i
i
⎝ i
⎠
(19)
where the complex Mueller matrix L of the complete system is given by
L ≡ ∑ piV (i ) = ∑ pi (T (i ) ⊗ T (i )∗ ) ,
i
(20)
i
and the corresponding Mueller matrix M is
⎛
⎞
⎛
⎞
M ≡ ALA −1 = A⎜ ∑ piV (i ) ⎟ A −1 = A⎜ ∑ pi (T (i ) ⊗ T (i )∗ ) ⎟ A −1
⎝ i
⎠
⎝ i
⎠
(i )
(i )∗ −1
(i )
= ∑ pi { A(T ⊗ T )A } = ∑ pi N .
i
(21)
i
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Journal Optical Society of America A, 17, 328-334, 2000
This result is obtained from considering the optical system as composed of a set of parallel
elements, but it is worth pointing out that the same result could be obtained by considering
the system as an ensemble7,15, so that each realization “i”, characterized by a well-defined
Jones matrix T(i), occurs with a probability pi. We can therefore consider the optical system
as composed of such an ensemble, and will refer to ensemble averages by means of
x
e
≡ ∑ p i x (i ) ,
(22)
i
so that
m kl = n kl
e
k , l = 0,1,2,3 .
,
(23)
4. Eigenvalue conditions
From Eq.(20) we can clearly see that the elements of the complex Mueller matrix L
L = T (i ) ⊗ T (i )∗
e
⎡
⎢
⎢
=⎢
⎢
⎢
⎣
t 0 t 0∗
t 0 t 2∗
e
e
∗
2 0 e
∗
2 2 e
t 0 t1∗
t 0 t 3∗
e
e
∗
2 1 e
∗
2 3 e
t1t 0∗
t1t 2∗
e
e
∗
3 0 e
∗
3 2 e
tt
tt
tt
tt
tt
tt
t1t1∗ ⎤
e
⎥
t1t 3∗ ⎥
e
⎥,
∗
t 3t1 ⎥
e
∗
t 3t 3 ⎥
e⎦
(24)
can be suitably reordered to construct the matrix H 25 whose elements are
hkl ≡
1
t k t l∗
2
e
, k , l = 0,1,2,3 .
(25)
The relation between the elements of M and H can be written as1
H=
1 3
∑ mkl Ekl ,
4 k ,l = 0
(26)
where Ekl are the following sixteen Dirac matrices
E kl = σ k ⊗ σ l (k, l = 0, 1, 2, 3),
(27)
so that
mkl = Tr( Ekl H ) .
(28)
The Hermitian matrix H is, in fact, the complex correlation matrix of the variables tk.
A necessary and sufficient condition for a Hermitian matrix H to be a complex correlation
matrix is that H be positive-semidefinite. Four characteristic conditions are then obtained
from the fact that a Hermitian matrix H is positive-semidefinite if, and only if, its
eigenvalues λi are non-negative 1,4,26,27.
An alternative set of four conditions equivalent to the “Eigenvalue Conditions” λi ≥ 0, but
with simpler explicit algebraic expressions, is provided by the non-negativity of four nested
principal minors of H 28.
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Journal Optical Society of America A, 17, 328-334, 2000
To obtain these expressions of the inequalities, it is useful to write the hkl elements in terms
of statistical parameters
hkl ≡ µ kl σ k σ l ,
(29a)
where
σ k2 = hkk = t k t k∗
e
= tk
2
e
.
(29b)
We will denote the modulus and the argument of µkl as follows
ρkl ≡ |µkl|, βkl ≡ arg (µkl).
(29c)
Then, a complete set of necessary and sufficient conditions for H to be a complex
correlation matrix, equivalent to the set of Eigenvalue Conditions, can be expressed from
the non-negativity of four nested principal minors of H (“Principal Minors Conditions”):
h00 ≥ 0 ;
(30a)
1 ≥ ρ 01 ;
(30b)
2
2
2
1 + 2ρ01ρ12ρ02cos (β01 + β12 − β02 ) ≥ ρ01
+ ρ12
+ ρ02
;
(30c)
det( H ) = 1 + 2ρ12 ρ 23ρ13 cos (β12 + β 23 − β13 ) + 2ρ 02 ρ 23ρ 03 cos (β 02 + β 23 − β 03 )
+ 2ρ 01ρ13ρ 03 cos (β 01 + β13 − β 03 ) + 2ρ 01ρ12 ρ 02 cos (β 01 + β12 − β 02 )
− 2ρ 01ρ 02 ρ13ρ 23 cos (β 01 − β 02 + β13 − β 23 )
(30d)
− 2ρ 01ρ 03ρ12 ρ 23 cos (β 01 − β 03 + β12 + β 23 )
− 2ρ 02 ρ 03ρ12 ρ13 cos (β 02 − β 03 − β12 + β13 )
2
2
2
2
2
2
2
2
2
2
+ ρ 01
ρ 223 + ρ 02
ρ13
+ ρ 03
ρ12
− ρ 01
− ρ 02
− ρ 03
− ρ12
− ρ13
- ρ 223 ≥ 0 .
In fact, all the 15 principal minors of H are non-negative but the above four inequalities
provide a set of sufficient conditions.
5. Transmittance conditions
The Eigenvalue Conditions have been obtained from the construction of the Mueller matrix
as an average over pure Mueller matrices, in the sense of
M= N
e
= A(T (i ) ⊗ T (i )∗ ) A −1
e
= A T (i ) ⊗ T (i )∗
e
A −1 ,
(31)
where T(i) are 2x2 complex matrices, but without taking into account the conditions derived
from T(i) being passive Jones matrices. Thus, in order to complete the set of conditions for
a 4x4 real matrix to be a physical Mueller matrix (i. e., a Mueller matrix corresponding to
an ensemble average of passive pure Mueller matrices), it is necessary to consider the
conditions derived from the transmittance condition being satisfied by all the pure Mueller
matrices of the elements involved in the average.
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Journal Optical Society of America A, 17, 328-334, 2000
As a first step towards obtaining these conditions we consider the quantity
(
2
2
2
g f ≡ m00 + m01
+ m02
+ m03
)
1
.
2
(32)
To demonstrate that gf ≤ 1 as in the pure case, we define the vectors
(
)
(i ) ( i ) (i )
m f ≡ (m01 , m02 , m03 ) , n (fi ) ≡ n01
, n02 , n03 ,
T
T
(33)
which allow us to write
1
2
2
⎧3 ⎡
⎤ ⎫⎪
(i ) ⎪
g f = m00 + m f = ∑ pi n00
+ ⎨∑ ⎢∑ pi n0(il) ⎥ ⎬ ,
i
⎪⎩ l =1 ⎣ i
⎦ ⎪⎭
(
)
(34)
and
2
⎧⎪ 3 ⎡
⎫
(i ) ⎤ ⎪
p
n
⎨∑ ⎢∑ i 0l ⎥ ⎬
⎪⎩ l =1 ⎣ i
⎦ ⎪⎭
(
)
⎡
(i )
= ⎢∑ pi2 n01
⎣ i
1
2
1
⎤2
⎥
⎦
( ) + ∑ p (n ( ) ) + ∑ p (n ( ) ) + ∑ p p (n ( )n ( ) + n ( )n ( ) + n ( )n ( ) )
2
2
i
2
i
02
2
i
i
i
03
2
j
i
⎡
2
= ⎢∑ pi2 n (fi ) + ∑ p j n (f j )
jk
⎣ i
j
01
k
k
01
jk
j
02
k
02
j
03
k
03
1
( )
T
⎤2
⎥ =
⎦
( )
p k n (fk )
Thus
(
∑ p n( ) ≤ ∑ p
i
i
i
f
i
n (fi ) .
i
)
(i )
(i )
g f ≤ ∑ p i n 00
+ ∑ p i n (fi ) = ∑ p i n 00
+ n (fi ) .
i
(35)
i
(36)
i
But, recalling the starting hypothesis that the gain condition is satisfied for all the pure
Mueller matrices N(i)
(i )
n00
+ n (fi ) ≤ 1 ,
(37)
and taking into account Eq. (15) we finally obtain
g f ≤ 1.
(38)
This well-known result has been obtained and studied previously by Barakat2 by directly
applying the Transmittance Condition to the Mueller matrix M. Nevertheless, it is
important to observe that the calculations followed by us to obtain this result are based on
two main hypotheses: (1) the system is considered as an ensemble (the Mueller matrix is
given by an ensemble average of pure Mueller matrices) and (2) each single realization is a
passive system (i. e., its pure Mueller matrix satisfies the Transmittance Condition).
We now define the magnitude g r
(
2
2
g r ≡ m00 + m102 + m20
+ m30
)
1
2
,
(39)
and we define the vectors
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Journal Optical Society of America A, 17, 328-334, 2000
(
)
(i ) ( i )
m r ≡ (m10 , m20 , m30 ) , n (ri ) ≡ n10(i ) , n20
, n30 .
T
T
(40)
The same calculations done for gf lead to
⎧3 ⎡
2 ⎤⎫
g r = ∑ pi n00 + ⎨∑ ⎢∑ pi nl(0i ) ⎥ ⎬
i
⎦⎭
⎩ l =1 ⎣ i
( )
(i )
1
2
(
)
(i )
≤ ∑ pi n00
+ n (ri ) ,
(41)
i
but taking into account that Eq.(15) is satisfied for all the pure Mueller matrices N(i) we see
that n (ri ) = n (fi ) , which allows us to write
∑ p (n ( ) + n ( ) ) = ∑ p (n ( ) + n ( ) ) ≤ 1 ,
i
00
i
i
r
i
i
i
00
i
f
(42)
i
and, consequently
gr ≤ 1 .
(43)
Thus, the procedure followed to construct the general Mueller matrices as an ensemble
average of physically realizable pure Mueller matrices leads to the fact that they must
satisfy, not only the “Transmittance Condition”
(
2
2
2
g f = m 00 + m 01
+ m 02
+ m 03
)
1
≤1,
(44a)
≤ 1,
(44b)
2
but also the following condition
(
2
2
g r ≡ m00 + m102 + m 20
+ m30
)
1
2
which can be called the “Reverse Transmittance Condition” because for most, if not all,
Mueller matrices gr represent the maximum transmittance when incident and emerging
light beams are interchanged29,30. Nevertheless, this new condition arises from the
hypotheses indicated and not from imposing additional conditions for the resultant Mueller
matrix.
The expressions of both Transmittance Conditions in function of the elements of H are the
following
[
= Tr (H ) + [(h
]
)]
∗
∗
)
g f = Tr (H ) + (h00 − h11 − h22 + h33 ) + 4(h01 + h23 ) (h01
+ h23
gr
2
(
∗
+ h11 − h22 − h33 ) + 4(h02 + h13 ) h02
+ h13∗
2
00
1
1
2
2
≤ 1,
(45a)
≤ 1.
(45b)
A good example for analyzing these properties of physical Mueller matrices is the
following matrix MLILP associated with a lossless ideal linear polarizer
M LILP
⎡1
⎢1
≡⎢
⎢0
⎢
⎣0
0
0
0
0
0
0
0
0
0⎤
0⎥⎥
.
0⎥
⎥
0⎦
(46)
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Journal Optical Society of America A, 17, 328-334, 2000
It is easy to check that this matrix satisfies the Eigenvalue Conditions as well as the
Transmittance Condition, but it does not satisfy the reverse transmittance condition. This is
consistent with the fact that this matrix can not be obtained as an average of pure Mueller
matrices.
6. Characterization theorem
The above results show that the Eigenvalue conditions and the two Transmittance
Conditions constitute a set of necessary conditions for a real 4x4 matrix to be a physical
Mueller matrix. Now we will show that these conditions are also sufficient.
Let us consider a real 4x4 matrix B which satisfies the Eigenvalue Conditions and the two
Transmittance Conditions. Now, by means of Eq. (26) we construct the corresponding
Hermitian positive-semidefinite matrix H, and obtain its spectral decomposition1
H = WΛW + = λ 0WD(1,0,0,0)W + + λ 1WD(0,1,0,0)W +
(47)
+ λ 2WD(0,0,1,0)W + + λ 3WD(0,0,0,1)W ,
where W is the unitary matrix that diagonalizes H, and λk are the eigenvalues of H.
By using Eq (28) to translate this result to matrix B, we obtain
3
[
k =0
]
3
λk
Tr ( H ) B ( k ) = ∑ p k N ( k ) ,
k = 0 Tr ( H )
k =0
3
B = ∑λ k B ( k ) = ∑
(48)
where
pk ≡
λk
,
Tr ( H )
3
∑p
k =0
k
=1 ,
[
]
N (k ) ≡ Tr (H )B (k ) ,
(k = 0,1,2,3) ;
(49)
so that B is written in Eq. (48) as an average of the four pure Mueller matrices N(k)
corresponding respectively to the four elements of the sum in Eq.(47).
To prove that N(k) satisfy the Transmittance Conditions we will take into account the
hypothesis
b00 + b f ≤ 1 ,
(50)
as well as the following equalities derived from the definition of the matrices B, B(k) and
N(k)
b00 = Tr ( H ) ,
b00(k ) = 1 ,
(k )
n00
= Tr ( H ),
(k = 0,1,2,3).
(51)
The vector
3
b f = ∑ p k n (fk )
(52)
k =0
is a convex linear combination of the four vectors n (fk ) . Now we note that all convex linear
combinations of n (fk ) correspond to points placed on the faces of a tetrahedron defined by
these vectors in the three-dimensional space. Thus, from Eq. (51), we see that the modulus
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Journal Optical Society of America A, 17, 328-334, 2000
of the convex linear combinations of n (fk ) are limited by (1-b00), and consequently the
modulus of these vectors are also limited by (1-b00)
n (fk ) ≤ 1 − b00 ,
(k = 0,1,2,3),
(53)
and, from Eq. (51) we can write
(k )
n (fk ) ≤ 1 − n00
,
(k = 0,1,2,3),
(54)
which is the Transmittance Condition for the four pure components N(k) of the matrix B.
From the above results we are ready to identify a complete set of necessary and sufficient
conditions for a real 4x4 matrix to be a physical Mueller matrix: (1) the Eigenvalue
Conditions [or, equivalently, the Principal Minors Conditions, given by Eqs.(29-30)] and
(2) the two Transmittance Conditions [given by Eq.(44) in terms of the elements of the
Mueller matrix M, or by Eq.(45) in terms of the elements of H(M)].
Thus we can state the following characterization theorem:
“A real 4x4 matrix M is a physical Mueller matrix if, and only if, the eigenvalues of H(M)
are non-negative and it satisfies the two Transmittance Conditions [Eq.(44)]” or, “A real
4x4 matrix M is a physical Mueller matrix if, and only if, the elements of H(M) satisfy the
Principal Minors Conditions [Eq.(30)] and the two Transmittance Conditions [Eq.(45)]”.
When a Mueller matrix is obtained from an experimental measurement, we of course
expect that, leaving aside experimental errors, it corresponds to a real optical system.
However, it is of interest to consider the above general constraints in order to know the
range of possible expectations as well as if the results obtained are consistent.
7. Purity criterion
The study of the necessary and sufficient conditions for a Mueller matrix to be a pure
Mueller matrix is an important subject dealt with by several authors.8-10,25,31-,37 We can now
analyze this subject in the light of the properties of the matrix H.
First we remember that Fry and Kattawar32 showed that the inequality
(
)
2
Tr M T M ≤ 4m00
(55)
is a necessary condition satisfied by any Mueller matrix, which yields equality for pure
Mueller matrices. However, the question whether, given a Mueller matrix M, the equality
(“Trace Condition”)
2
Tr (M T M ) = 4m00
(56)
is a sufficient condition for M to be a pure Mueller matrix has been a subject discussed in
some papers.8-10,26,38
At this point we will take the following previously presented statement8 (“Purity
Criterion”):
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Journal Optical Society of America A, 17, 328-334, 2000
“A necessary and sufficient condition for a Mueller matrix M to be a pure Mueller matrix is
2
that M satisfies the equality Tr (M T M ) = 4m00
”.
Here we include a demonstration based on the fact that pure Mueller matrices correspond to
Hermitian H matrices with only one non-zero eigenvalue (λ 0 ≠ 0, λ1 = λ 2 = λ 3 = 0) .
First let us note that demonstrating the following equalities is straightforward9
3
Tr ( M T M ) = 4Tr ( H 2 ) = 4∑ λ i2 ,
(57)
i =0
2
⎤
⎡ 3
2
2
m00
= (TrH ) = ⎢∑ λ i ⎥ .
⎣ i =0 ⎦
(58)
Taking into account that λi are non-negative, we can write
2
⎤
⎡ 3
λ
≤
∑
⎢∑ λ i ⎥ ,
i =0
⎣ i =0 ⎦
3
2
i
(59)
which can be expressed as
1
2
Tr ( M T M ) ≤ m00
.
4
(60)
We see that the equality is satisfied if, and only if, only one eigenvalue is non-zero, i.e. M is
a pure Mueller matrix.
To complete the demonstration, we note that the Transmittance Conditions are directly
satisfied from the starting hypothesis that M is a Mueller matrix.
The Purity Criterion is then valid for physical Mueller matrices (which satisfy the
Eigenvalue Conditions). Some works39,40 identify the set of Mueller matrices with the set
of “Stokes matrices” characterized by the fact that, for any Stokes vector the transformed
vector is also a Stokes vector. It is clear that any Mueller matrix is a Stokes matrix,
nevertheless the converse is not true4. In general, Stokes matrices don’t satisfy the
Eigenvalue Conditions4 and, consequently, the Purity Criterion is not applicable for these
matrices9.
Finally, we can use the Purity Criterion to define a nondimensional parameter for
measuring the “Degree of Purity” P(M) of an optical system represented by a Mueller
matrix M41.
2
⎡ Tr ( M T M ) − m00
⎤
P (M ) ≡ ⎢
⎥
2
3m00
⎣
⎦
12
12
⎧⎪ 1 ⎡ 4Tr ( H 2 ) ⎤ ⎫⎪
=⎨ ⎢
− 1⎥ ⎬ .
2
⎪⎩ 3 ⎣ (TrH )
⎦ ⎪⎭
(61)
The value of P(M) is restricted by 0 ≤ P(M ) ≤ 1 .
The limit P(M) = 0 corresponds to a total depolarizer whose Mueller matrix elements are
zero except m00 (i. e. to an ideal total depolarizer), and P(M) = 1 corresponds to a pure
Mueller matrix.
12
Journal Optical Society of America A, 17, 328-334, 2000
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