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
In this paper, we consider a new technique, depolarization synthesis, which stands in contrast to most other treatments of depolarization in optics , which deal mainly with analysis problems, i.e., to take a given depolarizing Mueller matrix 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 (where is an diagonal matrix) and depolarizes all incident light (called a perfect depolarizer ). 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 . This approach has recently been embedded in a symmetric product decomposition method  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 [1–3,9]. 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 ), 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 , we choose instead to define this process from the associated scattering coherency matrix , 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):
We then transform this vector using a unitary matrix as shown in Eq. (2):
Finally, we then form a Hermitian matrix from the elements of as shown in Eq. (4). Here, , where . Depolarizers are then formally defined as all systems where . This follows since the eigenvectors of (the columns of the unitary matrix ) are the Pauli matrix expansion coefficients of Jones matrices and hence it is only when one eigenvalue remains greater than zero that and hence corresponds to a pure Mueller–Jones matrix, see [1,9] and Section 2.3 of .
Conventionally, interest has centred on analysis of depolarizers, i.e., given a Mueller matrix , decompose the corresponding into its eigenvalues and eigenvectors [1–3]. However, here we consider the inverse process, namely depolarization synthesis. In this case we are not given , but instead are given a reference pure optical system (i.e., one with ). 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 Jones matrix and its Pauli expansion vector as shown in Eq. (5):7):
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
Depolarization synthesis can then be formally written as
The real triplet , , must lie inside a cube of dimension in order to keep the dominant Jones matrix . However, for each given triplet we then face a wide selection of depolarizers depending on the choice of the three orthogonal vectors , , and . We now turn to consider an efficient parameterization of these vectors.
The first key idea is that they must always be orthogonal to (since 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 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 ). This has application for example in backscatter problems when the reciprocity theorem forces in Eq. (6) .
C. Backscatter Case
The first idea is then to generate a unitary matrix , which transforms this vector into the unit vector corresponding to the identity matrix, i.e., to find such that
We can always find such a transformation as a product of elementary plane transformations, as shown in Eq. (12):
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 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 . The two states and then represent a reference frame orthogonal to . For example, in the important backscatter case of reflection symmetry, in Eq. (10) above (see ), and it follows that we can always rotate the matrix by an angle 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 ) together with cross polarization as a reference basis:
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 we can fill the whole matrix. For this reason we call the system unitary matrix.
Of course in general there can also be degrees of freedom in this , 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).
Here, is a null column vector (of dimension 2 in this case) and is a unitary matrix. Such matrices have two degrees of freedom, angles and , as shown in Eq. (16):
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 unitary matrix would include right multiplication of Eq. (16) by a phase matrix , but when we embed such components in the eigenvalue expansion of , 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 we can then generate a set of valid depolarizing unitary matrices , which have only two free parameters as shown in (17):
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 , and a unitary matrix , as shown in (18). These equations allow us to synthesize any rank-3 depolarizer built around a system using just two angles, and , and a real pair , . For any fixed value of these parameters we can then generate a matrix and the corresponding Mueller matrix 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:
D. General Depolarizer
Returning now to the general case of Eq. (6), the first stage is again to generate a unitary matrix , which transforms the vector into the unit vector corresponding to the identity matrix, i.e.,
We can always find such a transformation as a product of four elementary plane transformations, as shown in Eq. (20):
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 , , 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 unitary matrices. A convenient form is to employ a product decomposition of into a set of elementary plane rotations, as shown in Eq. (23) . 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 matrix used in backscatter problems:
Note for completeness that a general unitary matrix also has a diagonal phase shift matrix right multiplying in Eq. (23). This adds three degrees of freedom, but if we then further consider special unitary matrices () we obtain only two independent phase angles in 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 for a cascaded product involving and its conjugate transpose, the matrix factor 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 shown in Eq. (23).
Our final result, depolarization synthesis for arbitrary system geometry, is shown in Eq. (24). Here again we assume we know and and we then obtain nine parameters for synthesizing arbitrary depolarization for that system, given by a triplet of real eigenvalues , , 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 , , as axes. If we consider without loss of generalization that , 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 , , . 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 . 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:
Here, is the system unitary matrix and is the same for all optical systems, representing the depolarization transformations. Each of these two-dimensional factors of 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, and two of the three spheres become “fixed.” In this way we are limited to the geometry of a unit square in the , 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):
Here, depolarization depends only on the triplet , , . The angles and are properties of the “polarized” component of the scattering, which depends on phase angle, particle size, shape, and material composition . There are no depolarization parameters inside the matrix . 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 , as shown in Eq. (26):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 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:
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 Jones matrix as shown in Eq. (28):
Here, the first column of represents the Jones vector for the desired elliptical polarization. If we convert this matrix into its four-element vector we obtain a physical interpretation of the general vector from Eq. (6):
Note that if we are interested in backscatter polarizers, the reciprocity theorem forces , 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 and . These have a vector of the form . 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 ) or in man-made structures (Bragg scattering in liquid crystals ). 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):
Note that the two angles and 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 and hence derive a Mueller matrix using Eq. (2). These three component Mueller matrices are given as shown in Eq. (31):
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:
- (2) An interesting special case arises for . 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 or , 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 or to be close to zero due to the inherent handedness of the system. For example, the scarab beetle cuticle generates left circular polarization ( is close to zero) due to the natural handedness of biological Bouligand structures (chitin-based self-assembled multilayer helical stacks) . It has also been found experimentally to show depolarization behavior consistent with . However, in either case of handedness we see that a generic way in which the system can still depolarize is by , which is physically interpreted as a plane mirror reflection. Such a model was adopted in  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:
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 . The key point for us is that it yields a pure system, with a vector of the form shown in Eq. (34):
Note that to a good approximation is real and the normalized scattering depends only on a single parameter . Note also that normal incidence (for arbitrary dielectric constant) corresponds to , 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 and are depolarization eigenvector parameters and in this case is the underlying pure system parameter:
From these we again obtain a triplet of Mueller matrices as shown in (36):
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 . In this case, the above matrices simplify, as shown in Eq. (37):38). If we consider the special case , we obtain a particularly simple result, a diagonal depolarizer of the form shown in Eq. (39). This corresponds directly to a matrix of the form shown in Eq. (40), i.e., the eigenvectors are the Pauli matrices themselves so is associated with dihedral multiple scattering, while is the ideal mirror reflection and is cross polarization, and each of these basic processes are independent:
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 matrix.
To illustrate, and also to show that not all physical depolarizing systems are generated by depolarizing angles , , we choose an example from the literature . 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 :
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 , 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:
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 we obtain the estimated reciprocal backscatter matrix as shown in Eq. (43):
Here, we also show the eigenvalues of (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 , space. If we then extract the maximum eigenvector of we obtain the 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:
To analyze further the structure of depolarization of this system, we now use to generate a unitary transformation matrix and apply to the eigenvectors and phase normalize the unitary submatrix to obtain the following reduced matrix:
From this we see that this depolarizer has two nonzero angles and . It represents a specific type of depolarizer based on , but as we have shown in this paper, it is but one of a whole family of such depolarizers defined for this reference system at the , point. We could if we wish, then use Eq. (18) to study the general form of such a family.
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 . This has implications for the analysis of inverse problems, i.e., those problems where 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.
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