Abstract

In this paper, we consider the forward problem in depolarization by optical systems. That is, we seek a compact parameterization that allows us to take an arbitrary “pure” optical system (namely one defined by a single Mueller–Jones matrix) and model all possible ways in which that system can depolarize light. We model this structure using compound unitary transformations and illustrate physical interpretation of the parameters involved by considering four examples, the family of depolarizers generated by scattering by random nonspherical particle clouds. We then turn attention to circular polarizers before considering all ways in which mirror reflection can cause depolarization. Finally, we consider a numerical example applied to a published Mueller matrix for backscatter from chiral turbid media.

© 2013 Optical Society of America

1. INTRODUCTION

In this paper, we consider a new technique, depolarization synthesis, which stands in contrast to most other treatments of depolarization in optics [1], which deal mainly with analysis problems, i.e., to take a given depolarizing Mueller matrix [M] and break it down into components. This new approach has the advantage that we will be able to see exactly how a polarizing system (represented by a Mueller–Jones matrix) can change in form when depolarization is present. Importantly, we will be able to model all possible forms of depolarization or loss of coherence of the system.

The classic depolarizer has a simple Mueller matrix of [MD]=diag(m00000) (where diag(x1x2xN) is an N×N diagonal matrix) and depolarizes all incident light (called a perfect depolarizer [1]). At the other extreme we have “pure” Mueller matrices or Mueller–Jones matrices [2,3] which, while they may increase or decrease the degree of polarization of specific states, always preserve the degree of polarization of polarized light as unity, i.e., they do not depolarize polarized light. Nature provides of course a full spectrum of behaviors between these limits and it remains an unsolved problem how to parameterize fully the transition between these two extremes. In this paper, we attempt such a parameterization.

There have been two key stages in the development of our understanding of depolarizers. The first was to include a depolarizing matrix component in a product decomposition based on the polar decomposition of Jones matrices [4,5]. This polar approach, originally applied only to pure optical systems, yields a cascade of diattenuator and retarder and it was therefore convenient when the method was extended to maintain both pure retarder and diattenuator elements but to yield a remainder matrix, which is termed a depolarizer.

The second key idea was to consider the behavior of depolarizers under matrix transformations following application of the reciprocity theorem which leads to interest in diagonal depolarizers, which as their name suggest have a diagonal Mueller matrix given as a generalization of the perfect case so [MD]=diag(m00m11m22m33). This approach has recently been embedded in a symmetric product decomposition method [6] where the depolarization of any system is located in a triple matrix product of diagonal depolarizer sandwiched between two pure Mueller systems. Here it was found that while many systems have such a diagonal depolarization structure, some (called type 2 depolarizers) have a nondiagonal depolarizer component of a very particular form [7,8].

The limitation with both these approaches is that they are “analysis” driven, i.e., they follow from certain matrix factorization theorems (the singular value decomposition or SVD for the symmetric diagonal model and a polar decomposition of complex matrices for the first approach), which permits analysis of experimental matrices. However, they do not treat depolarization as a primary phenomenon but only as a “remainder” following estimation of pure components in a product.

In this paper, we put the emphasis from the start on depolarization and how it can occur in a general sense. The paper is therefore structured as follows. First, we define depolarization formally using the coherency matrix approach [13,9]. We then define the full set of reference pure systems within this algebra and model how such pure “states” can be depolarized using unitary matrix transformations. We develop backscatter optics as a special case and then consider the full theory before looking in detail at three applications, namely depolarization by scattering from random particle clouds [2,3], then the full family of depolarizers generated from a circular polarizer, and third, depolarizing mirrors and how they can be parameterized. We conclude with a numerical example applied to an experimentally measured Mueller matrix taken from the literature and then summarize the main results of our approach and suggest implications for the future treatment of depolarization problems in optics.

2. DEFINITION OF DEPOLARIZATION

We start by defining mathematically what we mean by depolarization. Although wave depolarization can have many different causes, both temporal and spatial (see [10]), here we treat its manifestation in the matrices dealing with the transformation of polarization state from point to point. Physically then it corresponds to a loss of coherence for polarized light incident on a system described by the matrix. While there are several ways to represent this idea algebraically from the Mueller matrix [M], we choose instead to define this process from the associated 4×4 scattering coherency matrix [T], which has been fully described for optical systems in [2,3,9,11]. To define this matrix we start by vectorizing the Mueller matrix itself as shown in Eq. (1):

[M]=[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]m̲=[m00m10m33].

We then transform this vector using a 16×16 unitary matrix [Q] as shown in Eq. (2):

t̲=[Q]m̲m̲=[Q]1t̲.

The matrix [Q] was first derived in [9] and has the explicit form shown in Eq. (3):

[Q]=12[Q1Q2Q3Q4Q2Q1iQ4iQ3Q3iQ4Q1iQ2Q4iQ3iQ2Q1][Q]1=[Q]*TQ1=[1000010000100001]Q2=[01001000000i00i0]Q3=[0010000i10000i00]Q4=[000100i00i001000].

Finally, we then form a 4×4 Hermitian matrix [T] from the elements of t̲ as shown in Eq. (4). Here, [D]=diag(λ1,λ2,λ3,λ4), where λ1λ2λ3λ40. Depolarizers are then formally defined as all systems where λ2>0. This follows since the eigenvectors of [T] (the columns of the 4×4 unitary matrix [U4]) are the Pauli matrix expansion coefficients of Jones matrices and hence it is only when one eigenvalue remains greater than zero that [T] and hence [M] corresponds to a pure Mueller–Jones matrix, see [1,9] and Section 2.3 of [11].

t̲=[t00t10t33][T]=[t00t01t02t03t01*t11t12t13t02*t12*t22t23t03*t13*t23*t33]=[U4][D][U4]*T.

Conventionally, interest has centred on analysis of depolarizers, i.e., given a Mueller matrix [M], decompose the corresponding [T] into its eigenvalues and eigenvectors [13]. However, here we consider the inverse process, namely depolarization synthesis. In this case we are not given [M], but instead are given a reference pure optical system (i.e., one with λ2=0). We then seek to parameterize the full set of all possible physical depolarizers based on this system. The first step in this process is to define a general reference pure system.

A. Reference Pure Optical System

We start by considering an arbitrary 2×2 Jones matrix [J] and its Pauli expansion vector k̲ as shown in Eq. (5):

[J]=[j11j12j21j22]=k0[1001]+k1[1001]+k2[0110]+k3[0ii0]kiC.

This C4 vector k̲ can then be further parameterized into unitary components as shown in Eq. (6) [8,9]:

k̲=[k0k1k2k3]=a[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4]=ae̲e̲*Te̲=1,
and finally by inverting this model we obtain a parameterization of the Jones matrix itself in terms of the Pauli unitary angles as shown in (7):
[J]=[j11j12j21j22]j11=a(cosαeiδ1+cosβsinαeiδ2),j12=asinβsinα(cosγeiδ3+isinγeiδ4),j21=asinβsinα(cosγeiδ3isinγeiδ4),j22=a(cosαeiδ1cosβsinαeiδ2).

This is our starting point. Any nondepolarizing system can be parameterized using Eq. (7). We now turn to consider how such systems may be depolarized.

B. Depolarization of the Reference System

We start with a reference k̲=ae̲1, which has a corresponding Mueller–Jones matrix [M1] and a coherency matrix [see Eq. (4)] [T1], written as an outer product as shown in Eq. (8) [8,9,11], where λ1=|a|2:

[T1]=λ1e̲1e̲1*T.

Depolarization synthesis can then be formally written as

[TD]=λ1e̲1e̲1*T+(λ2e̲2e̲2*T+λ3e̲3e̲3*T+λ4e̲4e̲4*T).

The real triplet λ2, λ3, λ4 must lie inside a cube of dimension λ1 in order to keep the dominant Jones matrix e̲1. However, for each given triplet we then face a wide selection of depolarizers depending on the choice of the three orthogonal vectors e̲2, e̲3, and e̲4. We now turn to consider an efficient parameterization of these vectors.

The first key idea is that they must always be orthogonal to e̲1 (since [TD] is Hermitian) and also be mutually orthogonal unit vectors. Our approach will then be to construct a reference basis set entirely from the first eigenvector e̲1 and then define a set of angles that allow rotations away from this reference set while maintaining all the orthogonalities required. To illustrate the methodology we begin with the three-dimensional case (when k̲C3). This has application for example in backscatter problems when the reciprocity theorem forces γ=π/2 in Eq. (6) [8].

C. Backscatter Case

In this case, one eigenvalue of [T] is always zero [2,11] and the reference eigenvector can be concatenated to three dimensions and has the general form shown in Eq. (10):

k̲=[k0k1k2]=λ1[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]=λ1e̲1.

The first idea is then to generate a 3×3 unitary matrix [U3R], which transforms this vector into the unit vector corresponding to the identity matrix, i.e., to find [U3R] such that

[U3R]e̲1=[100].

We can always find such a transformation as a product of elementary plane transformations, as shown in Eq. (12):

[U3R]=[cosαsinα0sinαcosα0001][1000cosβsinβ0sinβcosβ]·[eiδ1000eiδ2000eiδ3].

There are two important consequences of this factorization:

(1) By expanding the product in Eq. (12) and forming the conjugate transpose we obtain a reference basis set for the full C3 space, as in Eq. (13). The columns of this matrix then represent a reference set of pure systems, all defined from the known reference state e̲1. The two states e̲2r and e̲3r then represent a reference frame orthogonal to e̲1. For example, in the important backscatter case of reflection symmetry, δ2=δ3 in Eq. (10) above (see [11]), and it follows that we can always rotate the matrix by an angle β to obtain a canonical form, as shown in Eq. (14). Here we have two orthogonal scattering mechanisms (which in radar imaging may be dielectric surface and dihedral scattering for example [8]) together with cross polarization as a reference basis:

[U3R]=[e̲1*Te̲2r*Te̲3r*T][U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10cosβsinαeiδ2cosβcosαeiδ2sinβeiδ2sinβsinαeiδ3sinβcosαeiδ3cosβeiδ3],
[U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10sinαeiδ2cosαeiδ20001].

The main point, however, is that the whole frame is defined by the parameters of the dominant eigenvector. Once we know the first column of [U3R] we can fill the whole matrix. For this reason we call [U3R] the system unitary matrix.

Of course in general there can also be degrees of freedom in this e̲2r, e̲3r plane to rotate into new valid bases for synthesis. To see how we may parameterize such rotations we turn to the second important consequence of Eq. (12).

(2) Applying [U3R] to an arbitrary 3×3 unitary matrix with e̲1 as a first column [U3] provides a dimension reduction, as shown in Eq. (15) [12]:

[U3R][U3]=[10̲T0̲U2].

Here, 0̲ is a null column vector (of dimension 2 in this case) and U2 is a 2×2 unitary matrix. Such matrices have two degrees of freedom, angles σ and ϕ, as shown in Eq. (16):

U2(ϕ,σ)=[cosϕsinϕeiσsinϕeiσcosϕ]{0ϕπ2πσ<π.

We shall see that such component matrices are important in generalization of the theory to higher dimensions. Note that these two angles can also be interpreted geometrically as points on a sphere, a generalization of the Poincaré sphere for pure wave states [1,11]. Note also that a general 2×2 unitary matrix would include right multiplication of Eq. (16) by a phase matrix diag(eiθ1,eiθ2), but when we embed such components in the eigenvalue expansion of [T], such diagonal phase terms cancel and so they remain “hidden” parameters. For this reason they do not appear in Eq. (16).

Now we have fully parameterized the problem. With knowledge of the dominant eigenvector e̲1 we can then generate a set of valid depolarizing 3×3 unitary matrices [U3(ϕ,σ)], which have only two free parameters as shown in (17):

[U3(ϕ,σ)]=[U3R]*T[10̲T0̲U2(ϕ,σ)].

By explicit expansion of this matrix product we can then obtain our final result. Depolarization synthesis in the special case of backscatter can be formulated by a real pair λ2, λ3 and a 3×3 unitary matrix [U3(ϕ,σ)], as shown in (18). These equations allow us to synthesize any rank-3 depolarizer built around a system e̲1 using just two angles, ϕ and σ, and a real pair λ2, λ3. For any fixed value of these parameters we can then generate a matrix [TD] and the corresponding Mueller matrix [MD] using Eq. (2). In this way we can generate the full Mueller matrix family representing arbitrary depolarization of a given system. One nice feature of this algebraic approach is that it scales to arbitrary dimension and hence we now return to the general four-dimensional depolarization problem:

[TD]=[U3(ϕ,σ)][λ1000λ2000λ3][U3(ϕ,σ)]*Tλ1λ2,30R[U3(ϕ,σ)]=[e̲1e̲2e̲3]e̲1=[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]e̲2=[cosϕsinαeiδ1cosϕcosβcosαeiδ2sinϕsinβei(σ+δ2)cosϕsinβcosαeiδ3+sinϕcosβei(σ+δ3)]e̲3=[sinϕsinαei(δ1σ)cosϕsinβeiδ2sinϕcosβcosαei(δ2σ)cosϕcosβeiδ3sinϕsinβcosαei(δ3σ)].

D. General Depolarizer

Returning now to the general case of Eq. (6), the first stage is again to generate a 4×4 unitary matrix [U4R], which transforms the C4 vector into the unit vector corresponding to the identity matrix, i.e.,

[U4R]e̲1=[1000]T.

We can always find such a transformation as a product of four elementary plane transformations, as shown in Eq. (20):

[U4R]=[cosαsinα00sinαcosα0000100001][10000cosβsinβ00sinβcosβ00001]·[1000010000cosγsinγ00sinγcosγ][eiδ10000eiδ20000eiδ30000eiδ4].

By expanding this product we then obtain a reference basis for the synthesis of depolarizers, as shown in Eq. (21). Once again this reference frame is defined entirely from the first eigenvector. However, there remain degrees of freedom in the orthogonal subspace e̲2, e̲3, e̲4 to change the frame and maintain all the required orthogonalities. To find the degrees of freedom in the orthogonal depolarizing subspace we can again use the relation shown in Eq. (22), from which we see that the degrees of freedom are generated by a parameterization of 3×3 unitary matrices. A convenient form is to employ a product decomposition of [U3] into a set of elementary [U2] plane rotations, as shown in Eq. (23) [12]. Hence, we obtain a six-parameter representation of the depolarization subspace, conveniently represented in terms of three complex plane transformations, each one of which is similar to the [U2] matrix used in backscatter problems:

[U4R]=[e̲1*Te̲2*Te̲3*Te̲4*T][U4R]*T=[e̲1e̲2e̲3e̲4]e̲1=[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4],e̲2=[sinαeiδ1cosβcosαeiδ2cosγsinβcosαeiδ3sinγsinβcosαeiδ4]e̲3=[0sinβeiδ2cosγcosβeiδ3sinγcosβeiδ4],e̲4=[00sinγeiδ3cosγeiδ4],
[U4R][U4]=[10̲T0̲U3][U4]=[U4R]*T[10̲T0̲U3],
U3(ϕi,σi)=U23(ϕ3,σ3)U212(ϕ2,σ2)U13(ϕ1,σ1)U13=[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1]U12=[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]U23=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3].

Note for completeness that a general 3×3 unitary matrix also has a diagonal phase shift matrix [D3]=diag(eiθ1,eiθ2,eiθ3) right multiplying U23 in Eq. (23). This adds three degrees of freedom, but if we then further consider special unitary matrices (det(U3)=1) we obtain only two independent phase angles in [D3] and the eight degrees of freedom familiar from SU(3) representations [1,11,12]. However, since we are interested only in forming the Hermitian matrix [TD] for a cascaded product involving [U3] and its conjugate transpose, the matrix factor [D3] always cancels and so remains “hidden” in the construction of a depolarization subspace [8,9]. For this reason for polarization algebra we need use only a six-parameter representation of U3 shown in Eq. (23).

Our final result, depolarization synthesis for arbitrary system geometry, is shown in Eq. (24). Here again we assume we know e̲1 and λ1 and we then obtain nine parameters for synthesizing arbitrary depolarization for that system, given by a triplet of real eigenvalues λ2, λ3, λ4 and six angles, as shown in (24).

3. GEOMETRICAL MODEL FOR DEPOLARIZATION

Here we summarize the new approach to depolarization synthesis [Eq. (24)] and give a geometrical picture for the degrees of freedom involved. The starting point is to define a real three-dimensional space with λ2, λ3, λ4 as axes. If we consider without loss of generalization that λ1=1, then interest for depolarization is centred on the unit cube in the positive octant of this space. The origin of this space then represents the reference pure system, i.e., zero depolarization. All points inside the unit cube are then valid depolarization states of the system. Note that for this synthesis problem we drop the ordered requirement on the triplet λ2, λ3, λ4. In analysis it is always possible to reorder them in descending order, as required in Eq. (4), but here it is possible for example to have λ4>λ3. Indeed, it is possible to extend this to consider arbitrary order of all four eigenvalues. This would then correspond to selection not just of a dominant pure component as the reference but any process, even one associated with a minor eigenvalue. Here, however, for the sake of clarity, we maintain synthesis based on a dominant process only.

Now, at each point inside this unit cube we have hidden degrees of freedom. These may be considered a set of rotations of spheres. Equation (23) shows that an arbitrary three-dimensional unitary matrix can be expressed as a cascade of three two-dimensional unitary transformations:

[TD]=[U4(ϕi,σi)][λ10000λ20000λ30000λ4][U4(ϕi,σi)]*Tλ1λ2,3,40R,[U4(ϕi,σi)]=[e̲1e̲2e̲3e̲4]=[U4R]*T[10̲T0̲U3(ϕi,σi)],U3(ϕi,σi)=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3]·[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]·[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1].

Here, [U4R] is the system unitary matrix and [U3] is the same for all optical systems, representing the depolarization transformations. Each of these two-dimensional factors of [U3] has a geometrical interpretation in terms of the latitude and longitude of points on a sphere, one sphere for each plane component. Hence, we have a picture where, at each point of the unit cube, we have a cluster of three spheres, which may be independently oriented to reflect the degrees of freedom available for the synthesis of depolarizers with the same eigenvalue spectrum. Important special cases then arise. For example, in backscatter, λ4=0 and two of the three spheres become “fixed.” In this way we are limited to the geometry of a unit square in the λ2, λ3 plane, at each point of which we have a single rotating sphere generating all possible degrees of freedom for depolarization.

In a second important special case, scattering by random particle clouds [2,3,13], symmetry removes the spheres completely and we obtain a family of depolarizers generated by a conventional Euclidean space inside the unit cube as shown in Eq. (25):

[TD]=[U4(α,δ)][λ10000λ20000λ30000λ4][U4(α,δ)]*T[U4(α,δ)]=[cosαsinαeiδ00sinαeiδcosα0000100001].

Here, depolarization depends only on the triplet λ2, λ3, λ4. The angles α and δ are properties of the “polarized” component of the scattering, which depends on phase angle, particle size, shape, and material composition [11]. There are no depolarization parameters inside the matrix [U4]. Interestingly, we can then express all depolarization properties as an additive diagonal Mueller matrix. We do this by first rewriting the eigenvalue decomposition using an approach outlined in [12], as shown in Eq. (26):

[TD]=[U4(α,δ)][λ1λ2000000000000000][U4(α,δ)]*T+[λ20000λ20000λ30000λ4]=[T1]+[T2],
where the second matrix is independent of terms inside [U4], since it has equal eigenvalues in the e̲1, e̲2 subspace and so appears noise-like [1,14]. We can then convert each of the two factors to Mueller matrices using Eq. (2) to obtain Eq. (27). The first component here [M1] is a pure Mueller–Jones matrix, the process of depolarization is caused completely by the addition of the second diagonal matrix, which itself depends only on the eigenvalues as shown. However, important as this result is, we see it now as but a special case of the general depolarization synthesis of Eq. (24). To illustrate how depolarization is not always contained entirely in the eigenvalue spectrum, we consider two further examples. The first is generation of a family of depolarizers based on a circular polarizer as reference, and the second is the idea of backscatter from a depolarizing mirror:
[M]=[M1]+[MN]=m[1sin2αcosδ00sin2αcosδ10000cos2αsin2αsinδ00sin2αsinδcos2α]+[n000000n110000n220000n33],m=12(λ1λ2),n00=12(2λ2+λ3+λ4),n11=12(2λ2λ3λ4)n22=12(λ3λ4),n33=12(λ3λ4).

4. EXAMPLES: CIRCULAR DEPOLARIZERS AND DEPOLARIZING MIRRORS

Consider first a general polarizer. By definition, this is a pure optical system that generates the same (generally elliptical) polarization state for all incident waves. Such a system can be characterized by a (normalized) singular 2×2 Jones matrix as shown in Eq. (28):

[J]=[U2][D][U2]*T=[cosψsinψeiδsinψeiδcosψ][1000][cosψsinψeiδsinψeiδcosψ]=[cos2ψcosψsinψeiδcosψsinψeiδsin2ψ].

Here, the first column of [U2] represents the Jones vector for the desired elliptical polarization. If we convert this matrix into its four-element k̲ vector we obtain a physical interpretation of the general C4 vector from Eq. (6):

k̲=12[1cos2ψsin2ψcosδsin2ψsinδ]{α=45°δ1=δ2=δ3=δ4=0{β=ψγ=δ.

Note that if we are interested in backscatter polarizers, the reciprocity theorem forces δ=π/2, i.e., restriction to a set of polarization states lying along a great circle perpendicular to the equator of the Poincaré sphere and going through circular polarization at the poles. Let us then consider as an interesting special case circular polarizers defined by ψ=π/4 and δ=±π/2. These have a k̲ vector of the form [100±1]T. Now let us ask how to synthesize all possible depolarizers derived from such a pure system. Such circular polarizers arise both in nature (in the cuticle area of the scarab beetle Cetonia aurata for example [7]) or in man-made structures (Bragg scattering in liquid crystals [7]). In both cases we may depart from the perfectly pure system and obtain depolarization due to engineering or crystal imperfections or evolutionary structural limitations in biological systems. For the sake of simplicity in the algebra, let us consider backscatter only. In this case we can synthesize an arbitrary circular depolarizer by defining a triplet of eigenvectors using Eq. (18), as shown in Eq. (30):

[e̲1e̲2e̲3]=12[1cosϕsinϕeiσ02sinϕeiσ12cosϕ±1±cosϕsinϕeiσ].

Note that the two angles ϕ and σ here are depolarizing parameters. They represent ways in which the system can depart from ideal and still maintain its dominant purpose. From each column, we can then generate a coherency matrix [T]=e̲i·e̲i*T and hence derive a Mueller matrix [M] using Eq. (2). These three component Mueller matrices are given as shown in Eq. (31):

[M1]=[100±100000000±1001],[M2]=[112sin2ϕcosσ12sin2ϕsinσcos2ϕ12sin2ϕcosσsin2ϕ012sin2ϕcosσ12sin2ϕsinσ0sin2ϕ12sin2ϕsinσcos2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ],[M3]=[112sin2ϕcosσ12sin2ϕsinσsin2ϕ12sin2ϕcosσcos2ϕ012sin2ϕcosσ12sin2ϕsinσ0cos2ϕ12sin2ϕsinσsin2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ].

The general backscatter circular depolarizer can then be synthesized as a weighted sum of the three components, as shown in Eq. (32). Note two special features of this model:

  • (1) In all cases the reciprocity theorem is obeyed. [M] has the requisite symmetry and also it is easily confirmed that m00m11+m22m33=0 as required [2,11].
  • (2) An interesting special case arises for ϕ=0. In this case, the depolarizing circular polarizer takes the simplified form shown in Eq. (33). This is interesting for two reasons. First, it is an example of a so-called type 2 depolarizer in Mueller matrix theory (when λ1 or λ2=0, see discussion below) [7,8]. Here, we see it arises naturally as a consequence of orthogonality of components. Second, it has a simple physical interpretation in terms of three independent components, two pure circular polarizers of opposite sense and a pure mirror reflector. In practice we would expect one of λ1 or λ2 to be close to zero due to the inherent handedness of the system. For example, the scarab beetle cuticle generates left circular polarization (λ1 is close to zero) due to the natural handedness of biological Bouligand structures (chitin-based self-assembled multilayer helical stacks) [7]. It has also been found experimentally to show depolarization behavior consistent with ϕ=0 [7]. However, in either case of handedness we see that a generic way in which the system can still depolarize is by λ3, which is physically interpreted as a plane mirror reflection. Such a model was adopted in [7] to explain the observed experimental results but here we see that such a combination follows naturally from the algebra of the general depolarization synthesis problem:
    [M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=12sin2ϕcosσ(λ2λ3),m02=12sin2ϕsinσ(λ2λ3),m03=λ1λ2cos2ϕλ3sin2ϕ,m11=λ2sin2ϕ+λ3cos2ϕ,m12=0,m13=12sin2ϕcosσ(λ3λ2),m22=λ2sin2ϕλ3cos2ϕ,m23=12sin2ϕsinσ(λ2λ3),m33=λ1+cos2ϕ(λ2λ3),
    [M]=12[λ1+λ2+λ300λ1λ20λ30000λ30λ1λ200λ1+λ2λ3]=12[λ100λ100000000λ100λ1]+12[λ200λ200000000λ200λ2]+[λ30000λ30000λ30000λ3].

Staying with the idea of plane mirror reflection, we now turn to consider ways in which a mirror can itself depolarize. Again we restrict attention to backscatter since the algebra is easier (just two degrees of freedom in the depolarizing eigenvector space). However, rather than just consider normal incidence onto a reflecting surface, we generalize slightly to backscatter from a slightly rough surface at arbitrary angle of incidence. In this case, the solution is well known and given by the small-perturbation or Bragg model [11]. The key point for us is that it yields a pure system, with a k̲ vector of the form shown in Eq. (34):

k̲=[cosαsinα0].

Note that to a good approximation k̲ is real and the normalized scattering depends only on a single parameter α. Note also that normal incidence (for arbitrary dielectric constant) corresponds to α=π/2, when we obtain the pure Mueller matrix for plane mirror reflection used in Eq. (33).

The depolarization synthesis for this problem can then be formulated using Eq. (18) and again we first obtain a triplet of eigenvectors built from the dominant pure system. Again we remind the reader that ϕ and σ are depolarization eigenvector parameters and in this case α is the underlying pure system parameter:

[e̲1e̲2e̲3]=[cosαcosϕsinαsinϕsinαeiσsinαcosϕcosαsinϕcosαeiσ0sinϕeiσcosϕ].

From these we again obtain a triplet of Mueller matrices as shown in (36):

[M1]=[1sin2α00sin2α10000cos2α0000cos2α],[M2]=[1sin2αcos2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αcos2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσsin2ϕcos2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0sin2ϕcos2ϕcos2α],[M3]=[1sin2αsin2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αsin2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσcos2ϕsin2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0cos2ϕsin2ϕcos2α].

Again we can combine these three matrices weighted by an eigenvalue spectrum to obtain a complete synthesis of depolarization under Bragg backscattering. However, here we consider, as an example, the special case of a depolarizing mirror, i.e., the special case of normal incidence when α=π/2. In this case, the above matrices simplify, as shown in Eq. (37):

[M1]=[1000010000100001],[M2]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],[M3]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],
and so the general depolarizing mirror has a Mueller matrix of the form shown in Eq. (38). If we consider the special case ϕ=0, we obtain a particularly simple result, a diagonal depolarizer of the form shown in Eq. (39). This corresponds directly to a [T] matrix of the form shown in Eq. (40), i.e., the eigenvectors are the Pauli matrices themselves so λ2 is associated with dihedral multiple scattering, while λ1 is the ideal mirror reflection and λ3 is cross polarization, and each of these basic processes are independent:
[M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=m02=m13=m23=0,m03=sin2ϕcosσ(λ3λ2),m11=λ1+cos2ϕ(λ2λ3),m12=sin2ϕsinσ(λ3+λ2),m22=λ1+cos2ϕ(λ2λ3),m33=λ2+λ3λ1,
[M]=12[λ1+λ2+λ30000λ1+λ2λ30000λ2λ3λ10000λ2+λ3λ1],
[T]=[λ2000λ1000λ3].

A. Numerical Example

We have developed these ideas as a “synthesis” problem, i.e., to synthesize a complete family of depolarizers based on a reference “pure” system. However, we could also use this as an “analysis” technique, to determine which parameters define a given general [M] matrix.

To illustrate, and also to show that not all physical depolarizing systems are generated by depolarizing angles ϕ, σ=0, we choose an example from the literature [15]. In this case, we again choose backscatter (to make the algebra simpler). In principle we could use the same approach for any depolarizing system by employing higher dimensional unitary matrices as shown in Eq. (24).

We choose to employ an experimental matrix obtained for “backscatter” from a chiral turbid media (polystyrene microsphere suspension in a glucose aqueous solution). The Mueller matrix, calibrated and normalized to the 1, 1 element has the following published form [15]:

[M]=[1.0000.1150.0660.0230.1110.7590.0610.0010.0180.1510.4350.1390.0460.0060.1280.334].

This matrix, as it stands, does not satisfy the reciprocity theorem for backscatter and hence is probably measured close but not exactly in the backscatter direction. To see this we can form the coherency matrix [T], as shown in Eq. (42). Here, we see that the third row and column are much smaller than the others, indicating a close-to-backscatter measurement condition:

abs([T])=[0.4950.1750.0420.1070.1751.2640.0570.0240.0420.0570.0700.0060.1070.0240.0060.171][0.4950.17500.1070.1751.26400.02400000.1070.02400.171].

To form an estimate that satisfies the reciprocity theorem and has a rank-3 coherency matrix as required, we can null the third row and column as shown on the right-hand side. If we now convert back to [M] we obtain the estimated reciprocal backscatter matrix as shown in Eq. (43):

[M]=[1.0000.1170.0250.0120.1170.8230.1100.0030.0250.1100.4870.1380.0120.0030.1380.310]λ=1,0.378,0.104.

Here, we also show the eigenvalues of [T3] (normalized to the largest eigenvalue). We see that this system is indeed a depolarizer, coming from a class at the point 0.378, 0.104 in λ2, λ3 space. If we then extract the maximum eigenvector of [T3] we obtain the C3 unitary vector shown in Eq. (44). Here, we also show the Mueller–Jones matrix corresponding to this eigenvector. We see that it is close to that of a plane mirror reflector, but with important off-diagonal structure. This then represents our reference system:

e̲1=[0.2130.622i0.7530.020i0.008][M1]=[1.0000.2650.0190.0080.2650.9990.0040.0370.0190.0040.9090.3210.0080.0370.3210.908].

To analyze further the structure of depolarization of this system, we now use e̲1 to generate a unitary transformation matrix and apply to the eigenvectors and phase normalize the 2×2 unitary submatrix to obtain the following reduced matrix:

[U3]=[10000.9490.147i0.28000.147i0.2800.949]{ϕ=18.45°σ=62.30°.

From this we see that this depolarizer has two nonzero angles ϕ and σ. It represents a specific type of depolarizer based on [M1], but as we have shown in this paper, it is but one of a whole family of such depolarizers defined for this reference system [M1] at the λ2, λ3 point. We could if we wish, then use Eq. (18) to study the general form of such a family.

5. CONCLUSIONS

In this paper, we propose a new technique called depolarization synthesis. In this method we start from a known pure optical system with a given Jones matrix and from this derive a full parameterization of all possible depolarizers built from an associated system unitary matrix. To do this we first convert from the Mueller to the coherency matrix formulation and then employ two key ideas. The first, generation of a unitary matrix that transforms an arbitrary Jones matrix into the identity and the second, use of the same transformation to reduce the dimensionality of the eigenvector unitary matrix by one. This is then combined with a general product factorization of unitary matrices into plane transformations to obtain a full parameterization of depolarization. The final full set is then obtained by multiplication of the system unitary matrix by the depolarization matrix. This yields a parameterized set of eigenvectors which can then be weighted by eigenvalues and transformed back into the more familiar Mueller matrix form.

We first developed the idea for rank-3 Hermitian matrices, corresponding to the important case of backscatter, before generalizing the method to arbitrary optical depolarizing systems with a rank-4 coherency matrix. We note that the same approach could in future be scaled to higher dimensions to model depolarization in other physical problems. The procedure easily generalizes to arbitrary rank-N Hermitian matrices.

To illustrate we chose three applications. The first, scattering by random particle clouds, when the symmetry of the problem removes all unitary matrix factors from the depolarization problem and leaves only the eigenvalue spectrum. We then considered two applications where this is not the case, first for circular depolarizers and second for depolarizing mirrors. For both these we were able to find explicit equations for the arbitrary depolarizer and show how they relate to observations made in the literature. Finally, we applied the technique to analysis of an experimental Mueller matrix obtained for backscatter from a chiral turbid medium. Here we were able to demonstrate how physical depolarization processes may be fully characterized by the new formalism, and how it leads to the possibility of studying families of depolarizers related through physical connection to the same underlying “pure” optical system.

There are two important conclusions from this study. First, this approach enables isolation and study of all ways in which any optical system can depolarize and so will be of interest in studies of optical system design optimization and also in the interpretation of depolarization in remote sensing and Mueller matrix imaging applications. Second, our study shows that depolarization is not restricted to diagonal Mueller matrices and in general can influence all elements of [M]. This has implications for the analysis of inverse problems, i.e., those problems where [M] is given and one seeks to isolate the depolarization factors from other elements and use them to interpret physical structure within the system.

The main advantage of this new approach for remote sensing and image processing problems is the context it gives to other methods. For example, if you use SVD or polar approaches in your application, we have shown here that you are implicitly assuming specific forms for the nature of the depolarization, which may be wrong for the physics of your problem. Whether this is important or not can now be assessed by using synthesis techniques such as those outlined in this paper.

REFERENCES

1. J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007). [CrossRef]  

2. M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

3. J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

4. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]  

5. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007). [CrossRef]  

6. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009). [CrossRef]  

7. R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426–2428 (2009). [CrossRef]  

8. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010). [CrossRef]  

9. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

10. M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: electromagnetic prediction,” Opt. Express 18, 15832–15843 (2010). [CrossRef]  

11. S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

12. F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).

13. S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009). [CrossRef]  

14. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011). [CrossRef]  

15. S. Manhas, M. K. Swami, P. Buddhiwant, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  2. M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).
  3. J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.
  4. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  5. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
    [CrossRef]
  6. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  7. R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426–2428 (2009).
    [CrossRef]
  8. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
    [CrossRef]
  9. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).
  10. M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: electromagnetic prediction,” Opt. Express 18, 15832–15843 (2010).
    [CrossRef]
  11. S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).
  12. F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).
  13. S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
    [CrossRef]
  14. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  15. S. Manhas, M. K. Swami, P. Buddhiwant, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
    [CrossRef]

2011

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

2010

2009

2007

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

2006

1996

1986

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

Amra, C.

Buddhiwant, P.

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
[CrossRef]

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

De Martino, A.

Domke, H.

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Fallet, C.

Foldyna, M.

Gil, J. J.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

Gupta, P. K.

Guyot, S.

Hovenier, J. W.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Lu, S.-Y.

Manhas, S.

Mischenko, M. I.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

Murnaghan, F. D.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).

Ossikovski, R.

San José, I.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

Singh, K.

Soriano, G.

Sorrentini, J.

Swami, M. K.

Travis, L. D.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

van der Mee, C.

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Zerrad, M.

Eur. Phys. J. Appl. Phys.

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
[CrossRef]

Opt. Commun.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

Other

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (45)

Equations on this page are rendered with MathJax. Learn more.

[M]=[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]m̲=[m00m10m33].
t̲=[Q]m̲m̲=[Q]1t̲.
[Q]=12[Q1Q2Q3Q4Q2Q1iQ4iQ3Q3iQ4Q1iQ2Q4iQ3iQ2Q1][Q]1=[Q]*TQ1=[1000010000100001]Q2=[01001000000i00i0]Q3=[0010000i10000i00]Q4=[000100i00i001000].
t̲=[t00t10t33][T]=[t00t01t02t03t01*t11t12t13t02*t12*t22t23t03*t13*t23*t33]=[U4][D][U4]*T.
[J]=[j11j12j21j22]=k0[1001]+k1[1001]+k2[0110]+k3[0ii0]kiC.
k̲=[k0k1k2k3]=a[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4]=ae̲e̲*Te̲=1,
[J]=[j11j12j21j22]j11=a(cosαeiδ1+cosβsinαeiδ2),j12=asinβsinα(cosγeiδ3+isinγeiδ4),j21=asinβsinα(cosγeiδ3isinγeiδ4),j22=a(cosαeiδ1cosβsinαeiδ2).
[T1]=λ1e̲1e̲1*T.
[TD]=λ1e̲1e̲1*T+(λ2e̲2e̲2*T+λ3e̲3e̲3*T+λ4e̲4e̲4*T).
k̲=[k0k1k2]=λ1[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]=λ1e̲1.
[U3R]e̲1=[100].
[U3R]=[cosαsinα0sinαcosα0001][1000cosβsinβ0sinβcosβ]·[eiδ1000eiδ2000eiδ3].
[U3R]=[e̲1*Te̲2r*Te̲3r*T][U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10cosβsinαeiδ2cosβcosαeiδ2sinβeiδ2sinβsinαeiδ3sinβcosαeiδ3cosβeiδ3],
[U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10sinαeiδ2cosαeiδ20001].
[U3R][U3]=[10̲T0̲U2].
U2(ϕ,σ)=[cosϕsinϕeiσsinϕeiσcosϕ]{0ϕπ2πσ<π.
[U3(ϕ,σ)]=[U3R]*T[10̲T0̲U2(ϕ,σ)].
[TD]=[U3(ϕ,σ)][λ1000λ2000λ3][U3(ϕ,σ)]*Tλ1λ2,30R[U3(ϕ,σ)]=[e̲1e̲2e̲3]e̲1=[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]e̲2=[cosϕsinαeiδ1cosϕcosβcosαeiδ2sinϕsinβei(σ+δ2)cosϕsinβcosαeiδ3+sinϕcosβei(σ+δ3)]e̲3=[sinϕsinαei(δ1σ)cosϕsinβeiδ2sinϕcosβcosαei(δ2σ)cosϕcosβeiδ3sinϕsinβcosαei(δ3σ)].
[U4R]e̲1=[1000]T.
[U4R]=[cosαsinα00sinαcosα0000100001][10000cosβsinβ00sinβcosβ00001]·[1000010000cosγsinγ00sinγcosγ][eiδ10000eiδ20000eiδ30000eiδ4].
[U4R]=[e̲1*Te̲2*Te̲3*Te̲4*T][U4R]*T=[e̲1e̲2e̲3e̲4]e̲1=[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4],e̲2=[sinαeiδ1cosβcosαeiδ2cosγsinβcosαeiδ3sinγsinβcosαeiδ4]e̲3=[0sinβeiδ2cosγcosβeiδ3sinγcosβeiδ4],e̲4=[00sinγeiδ3cosγeiδ4],
[U4R][U4]=[10̲T0̲U3][U4]=[U4R]*T[10̲T0̲U3],
U3(ϕi,σi)=U23(ϕ3,σ3)U212(ϕ2,σ2)U13(ϕ1,σ1)U13=[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1]U12=[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]U23=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3].
[TD]=[U4(ϕi,σi)][λ10000λ20000λ30000λ4][U4(ϕi,σi)]*Tλ1λ2,3,40R,[U4(ϕi,σi)]=[e̲1e̲2e̲3e̲4]=[U4R]*T[10̲T0̲U3(ϕi,σi)],U3(ϕi,σi)=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3]·[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]·[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1].
[TD]=[U4(α,δ)][λ10000λ20000λ30000λ4][U4(α,δ)]*T[U4(α,δ)]=[cosαsinαeiδ00sinαeiδcosα0000100001].
[TD]=[U4(α,δ)][λ1λ2000000000000000][U4(α,δ)]*T+[λ20000λ20000λ30000λ4]=[T1]+[T2],
[M]=[M1]+[MN]=m[1sin2αcosδ00sin2αcosδ10000cos2αsin2αsinδ00sin2αsinδcos2α]+[n000000n110000n220000n33],m=12(λ1λ2),n00=12(2λ2+λ3+λ4),n11=12(2λ2λ3λ4)n22=12(λ3λ4),n33=12(λ3λ4).
[J]=[U2][D][U2]*T=[cosψsinψeiδsinψeiδcosψ][1000][cosψsinψeiδsinψeiδcosψ]=[cos2ψcosψsinψeiδcosψsinψeiδsin2ψ].
k̲=12[1cos2ψsin2ψcosδsin2ψsinδ]{α=45°δ1=δ2=δ3=δ4=0{β=ψγ=δ.
[e̲1e̲2e̲3]=12[1cosϕsinϕeiσ02sinϕeiσ12cosϕ±1±cosϕsinϕeiσ].
[M1]=[100±100000000±1001],[M2]=[112sin2ϕcosσ12sin2ϕsinσcos2ϕ12sin2ϕcosσsin2ϕ012sin2ϕcosσ12sin2ϕsinσ0sin2ϕ12sin2ϕsinσcos2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ],[M3]=[112sin2ϕcosσ12sin2ϕsinσsin2ϕ12sin2ϕcosσcos2ϕ012sin2ϕcosσ12sin2ϕsinσ0cos2ϕ12sin2ϕsinσsin2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ].
[M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=12sin2ϕcosσ(λ2λ3),m02=12sin2ϕsinσ(λ2λ3),m03=λ1λ2cos2ϕλ3sin2ϕ,m11=λ2sin2ϕ+λ3cos2ϕ,m12=0,m13=12sin2ϕcosσ(λ3λ2),m22=λ2sin2ϕλ3cos2ϕ,m23=12sin2ϕsinσ(λ2λ3),m33=λ1+cos2ϕ(λ2λ3),
[M]=12[λ1+λ2+λ300λ1λ20λ30000λ30λ1λ200λ1+λ2λ3]=12[λ100λ100000000λ100λ1]+12[λ200λ200000000λ200λ2]+[λ30000λ30000λ30000λ3].
k̲=[cosαsinα0].
[e̲1e̲2e̲3]=[cosαcosϕsinαsinϕsinαeiσsinαcosϕcosαsinϕcosαeiσ0sinϕeiσcosϕ].
[M1]=[1sin2α00sin2α10000cos2α0000cos2α],[M2]=[1sin2αcos2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αcos2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσsin2ϕcos2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0sin2ϕcos2ϕcos2α],[M3]=[1sin2αsin2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αsin2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσcos2ϕsin2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0cos2ϕsin2ϕcos2α].
[M1]=[1000010000100001],[M2]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],[M3]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],
[M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=m02=m13=m23=0,m03=sin2ϕcosσ(λ3λ2),m11=λ1+cos2ϕ(λ2λ3),m12=sin2ϕsinσ(λ3+λ2),m22=λ1+cos2ϕ(λ2λ3),m33=λ2+λ3λ1,
[M]=12[λ1+λ2+λ30000λ1+λ2λ30000λ2λ3λ10000λ2+λ3λ1],
[T]=[λ2000λ1000λ3].
[M]=[1.0000.1150.0660.0230.1110.7590.0610.0010.0180.1510.4350.1390.0460.0060.1280.334].
abs([T])=[0.4950.1750.0420.1070.1751.2640.0570.0240.0420.0570.0700.0060.1070.0240.0060.171][0.4950.17500.1070.1751.26400.02400000.1070.02400.171].
[M]=[1.0000.1170.0250.0120.1170.8230.1100.0030.0250.1100.4870.1380.0120.0030.1380.310]λ=1,0.378,0.104.
e̲1=[0.2130.622i0.7530.020i0.008][M1]=[1.0000.2650.0190.0080.2650.9990.0040.0370.0190.0040.9090.3210.0080.0370.3210.908].
[U3]=[10000.9490.147i0.28000.147i0.2800.949]{ϕ=18.45°σ=62.30°.

Metrics