Optics and Lasers in Engineering 172 (2024) 107830 Contents lists available at ScienceDirect Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng Connecting the microscopic depolarizing origin of samples with macroscopic measures of the Indices of Polarimetric Purity Mónica Canabal-Carbia a, *, Irene Estévez a, Esther Nabadda b, Enrique Garcia-Caurel c, J.J. Gil d, Razvigor Ossikovski c, Andrés Márquez e, f, Ignacio Moreno b, g, Juan Campos a, Angel Lizana a a Grup d’Òptica, Departamento de Física, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain b Instituto de Bioingeniería, Universidad Miguel Hernandez de Elche, Elche 03202, Spain c LPICM, CNRS, Ecole Polytechnique, Institut Politechnique de Paris, Palaiseau 91120, France d Departmento de Física Aplicada, Universidad de Zaragoza, Pedro Cerbuna 12, Zaragoza 50009, Spain e Departamento de Física, Ing. de Sistemas y T. Señal, Universidad de Alicante, Alicante 03080, Spain f I.U. Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Alicante 03080, Spain g Departamento de Ciencias de Materiales, Óptica y Tecnología Electrónica, Universidad Miguel Hernandez, Elche 03202, Spain A R T I C L E I N F O A B S T R A C T Keywords: In this work we show how a specific set of three depolarizing observables, the Indices of Polarimetric Purity Mueller matrices (IPP), P1, P2 and P3, are ideal metrics to study the depolarization characteristic of media. We simulate different Indices of Polarimetric Purity depolarizing scenarios, based on different depolarizing origins, and we study the corresponding IPP values. The Depolarizers simulations are based on the incoherent addition of multiple elemental polarizing elements, as ideal polarizers Biophotonics Polarization and/or retarders with different specific characteristics (orientation, retardance, transmittance, etc.). Further depolarizing scenarios are also studied by including the effect of ideal depolarizers. We show for the first time how by analyzing depolarizing systems through IPP we unravel two different depolarizing origins: isotropic and anisotropic depolarization, with meaningful physical interpretation. The former, isotropic depolarization is related to pure scattering processes, and mainly connected with P3 observable. The later, anisotropic depolari­ zation is originated by microscopic constituent elements showing polarimetric anisotropy (dichroic and/or birefringent elements with different characteristics) and anisotropic scattering produced by these elements, and mainly described by P1 and P2 observables. Both effects can be simultaneously observed in real samples and give us information of the processes that give rise to depolarization in light-matter interactions. The simulated results are experimentally validated by analyzing the depolarizing behavior, in terms of IPP, of diverse real samples with easy physical interpretation, and direct connection with simulations. The present study could be of interest in multiple scenarios, to further understand the depolarizing response of samples, and it can be of special interest for the study of biological tissues and pathologies, as they present important depolarizing behavior. 1. Introduction being of interest in terms of contrast enhancement between different structures in the samples. These differences in the exiting polarization Polarimetry encompasses a collection of optical techniques devoted after interacting with different tissues are related to the specific polar­ to analyzing one main characteristic of transversal waves, the polari­ imetric characteristics of tissues, as the birefringence, dichroism, or zation, or the polarimetric property of samples, through light-matter depolarization. Typically, birefringence and dichroism are two polari­ interactions. Nowadays, polarimetry is useful in a wide range of appli­ metric responses well-studied in biological samples, as it is common to cations, as in astronomy [1], remote sensing [2], material character­ find that constituent elements of tissues present some significant ization and quality control [3,4], food analysis [5], botanical behavior in terms of birefringence or dichroism. For instance, in Ref. [8] applications [6], biomedicine [7], among other. the birefringent properties of a sample help to differentiate between In the case of biological samples, polarized light may be modified in healthy and cancerous tissue, in Ref. [9] elastic and collagen fibers in a different ways when interacting with different organic structures, this rabbit aortic wall are characterized by its retardance, and in Ref. [10] * Corresponding author. E-mail address: monica.canabal@uab.cat (M. Canabal-Carbia). https://doi.org/10.1016/j.optlaseng.2023.107830 Received 29 May 2023; Received in revised form 5 September 2023; Accepted 6 September 2023 Available online 20 September 2023 0143-8166/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 diattenuation allows to distinguish between brain regions with different of the light is described by the Stokes vector (S), whereas the polari­ tissue properties. metric properties of samples are described by Mueller matrices (M) [17, Depolarization is a measure of the sample capability to depolarize an 28–32]. incident fully polarized light beam. Traditionally, in terms of image In the following, we briefly review the calculation of the IPP. These contrast, depolarization was seen as a non-desired magnitude, which three magnitudes arise naturally when applying the spectral theorem was tried to be minimized to exploit the other polarimetric properties. [33] on polarimetric systems. The IPP constitute a set of three real However, in the last decades, it has been revealed that depolarization magnitudes, named P1, P2 and P3, defined as combinations of the channel itself provides large intrinsic information of samples (as for covariance matrix H (associated with M) eigenvalues (λi) [17,20,33]: instance, organization, size, density of structures, etc.), as well as it is λ0 − λ1 very relevant in terms of structures visualization enhancement. In this P1 ≡ , TrH sense, depolarizing channels have been reported as useful to detect λ0 + λ1 − 2λ2 cancer stages in different human and animal tissues as ex-vivo human P2 ≡ , (1) TrH colon and skin [11,12] colon cancer, skin carcinoma, pathology detec­ λ0 + λ1 + λ2 − 3λ3 tion in vegetal tissue [13] and, tissue recognition [14]. P3 ≡ TrH , An ideal framework to study depolarizing properties of samples is the Mueller–Stokes (M-S) formalism. In this sense, the Mueller matrix of a where, the values are restricted between 0 and 1, and follow the relation: sample, can be experimentally determined from intensity measure­ 0 ≤ P1 ≤ P2 ≤ P3 ≤ 1, (i = 1, 2, 3) (2) ments. Different observables that can be deduced from the coefficients of the Mueller matrix allow to study the depolarizing properties of samples Pure or non-depolarizing systems are characterized by P1=P2=P3=1, [15–17]. Among these different approaches, in this work we use the while for totally or ideal depolarizers P1=P2=P3=0. Indices of Polarimetric Purity (IPP) [18–20] as reference metrics to Importantly, the characteristic decomposition of the Mueller matrix study the depolarization. The IPP comprises a set of three observables, can be expressed in terms of the IPP as [19,20]: the P1, P2 and P3, channels, that can be derived from the eigenvalues of ̂ J0 ) + (P2 − P1 )(m00 M M = P1 (m00 M ̂ 1 ) + (P3 − P2 )(m00 M ̂ 2) the covariance matrix [20] (a transformation of the Mueller matrix), and they are connected to the polarimetric randomness structure of the ̂ 3) + (1 − P3 )(m00 M (3) sample that they represent [21–23]. The IPP are an ideal framework to study the depolarizing properties of biological samples, as they have where, P1 is the weight of the nondepolarizing component M ̂ J0 , P2-P1 is already proved their interest for the image enhancement and discrimi­ the portion of the medium that behaves as a 2D depolarizer (represented nation of biological structures [6,13,24,25] as well as to construct by M̂ 1 ), P3-P2 summarizes the part of the medium that behaves as a 3D recognition models for the guided detection of biological structures depolarizer (represented by M̂ 2 ), 1-P3 is the portion of the medium [14]. behaving as a perfect depolarizer where M ̂ 3 =diag(1,0,0,0) and the In this work we take advantage of the IPP as a metrics to study the circumflex in M denotes the normalized matrix in each case [17,19,30]. depolarization origin. For this purpose, we construct different de­ Note how Eq. (3) becomes very interesting because it allows us polarizers through the incoherent summation of different basic elements connecting the weights of the different terms of the characteristic (pure diattenuators, pure retarders and perfect depolarizers), with decomposition, which are physically interpretable, with combinations different properties and orientations. The resulting simulated scenarios, of IPP, which can be experimentally obtained. studied in terms of IPP, allow us to understand the inherent constituents within samples with regards to their macroscopic depolarizing response. 3. Depolarizing systems with P3=1 In this sense, the proposed simulations allow us to differentiate between different kind of depolarizers, arising two main depolarizing mecha­ Depolarizing behavior of samples can be understood as their ability nisms: (1) depolarizers whose depolarizing response does not depend on to introduce polarimetric randomness to an input polarized light beam. the input state of polarization; and (2) depolarizers whose depolarizing When an expanded beam illuminates a sample, light-matter interactions response depends on the input polarization. Importantly, we demon­ are produced which result in heterogeneous modifications of the po­ strate how those depolarizing vias can be quantified by using the IPP larization spatial distribution of light. Those polarization spatial distri­ framework. In addition, the proposed simulated environments are butions are incoherently added at the detectors of polarimeters, leading experimentally validated with simple samples with well-defined polar­ to depolarization measurements. The physical origin behind the depo­ ization properties that reproduce phantom experiments. larizing response of a sample is multiple, but it is always connected to The presented study allows us to connect inherent physical proper­ the inherent constituents of the sample, and how they modify polari­ ties of samples with measurable depolarization data and it provides new zation through light-matter interactions. In this work, we provide analytical tools for a better understanding of depolarizing mechanisms different cases of study which lead to depolarization, and they are very in biological samples, and therefore, to their inherent structures. easy to interpret. The cases will be studied in terms of their Indices of Polarimetric Purity (IPP) values, because, as we will show, the IPP allow 2. Indices of Polarimetric Purity (IPP) as a framework to study us to catalog different kinds of depolarizers as a function of the inherent depolarizing behavior of samples processes that give rise to the depolarization. To do so, we will model different depolarizer scenarios in which In this section we provide the fundamentals necessary to construct depolarization takes place, consisting of basic physical elements, this the models given in further Section 3, which will help us to study leading to a very simple interpretation of the depolarizing processes different depolarizing scenarios. behind them. In particular, in this section we present two different In particular, we focus on the IPP to provide fundamental informa­ simulations of depolarizers that are based on the simplest polarimetric tion about the origin of depolarization. The choice of the IPP is not elements: linear retarders and linear diattenuators. As it will be shown, arbitrary since they provide further information than other well-known these two scenarios lead to the particular value of P3=1, whereas P1 and depolarizing observables (as the Degree of Polarization or the Depola­ P2 are different from case to case. In Section 1.3 of the supplementary rizing index [27]) as well as because their interest for the enhanced document, we also include the simulation where the Mueller matrix is visualization of tissues has already been demonstrated [13]. To study constructed by the combination of linear diattenuators and retarders. the IPP parameters we construct our simulations and experiments based The behavior of these M is similar to the ones presented in the main text, on the M-S formalism [28,29]. In this formalism, the state of polarization 2 M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 and P3=1 for all the cases. from 0 to 1 with steps of 0.003 and with a null variance (σpy=0), and The models presented in the following are implemented in a way that values of θ with a given mean value (we choose θ=60º and a variance is in adequacy with the premises of the parallel decomposition [17]. The (σθ) taking values from 0º to 180º with steps of 0.6º. This lead to a 300 x latter decomposition states that, any depolarizing M can be described as 300 = 90,000 simulations, each one leading to a Mueller matrix of a an incoherent sum of Mueller matrices Mi [18,19,21,33,34] composed depolarizer. In order to study the depolarizing properties of ∑ n ∑ n the resulting 90,000 depolarizers, the IPP (i.e., P1, P2 and P3 channels) ̂ = m00 M = m00 M ̂ i ; αi ≥ 0; αi M αi = 1 (4) values are calculated according to formulation given in Section 2. i i=1 The results obtained for P1, P2 and P3 are presented in Fig. 1. The x In this section, we conduct the simulations based on the incoherent axis provides the py parameter, whereas the y axis shows the values for addition of n Mueller matrices of pure polarimetric elements (linear the variance of θ, the σ θ parameter. Therefore, the resulting images diattenuators and linear retarders; with different values and orienta­ include the IPP values for the 90,000 simulations, each pixel of the tions). These simulations pretend to mimic the macroscopic response of image corresponding to a particular (py,i, σθ combination). The values for complex systems including dichroism, birefringence, or both. the IPP are coded in a color scale ranging from the minimum value (0 value, in blue) to the maximum value (1 value, in yellow), as provided by the colorbar. In addition, different rows in Fig. 1 provide the IPP 3.1. Depolarizers originated by the incoherent addition of linear simulations for two different values of px (0.8 and 0.2). We have also diattenuators computed the simulations for different values of px, these results are presented in Section 1 of the supplementary material. The first case of study consists of the simulation of depolarizers In Fig. 1(a)–(f) we observe the behavior of different depolarizers constituted by the incoherent addition of linear diattenuators. The based on incoherent addition of linear diattenuators with different Mueller matrix of a x-y aligned linear diattenuator MLD can be written as characteristics of (px, py, and σ θ). Regarding the orientation angle mean [28]: value for the constituent diattenuators, σθ, this parameter is set as 60º for ⎛ ⎞ 2 2 2 2 all simulations in Fig. 1. The choice of such angle was arbitrary as we ⎜ px + py px − py 0 0 ⎟ have observed that simulations are independent of this parameter. In ⎜ ⎟ ( ) 1 ⎜ p2x − p2y p2x + p2y 0 0 ⎟ fact, the orientation angle does not affect the polarimetric characteris­ MLD px , py = ⎜ ⎟; 0 ≤ px,y ≤ 1, (5) 2⎜⎜ 0 0 2px py 0 ⎟ ⎟ tics of the resulting depolarizers since it can be understood as a rotation ⎝ ⎠ 0 0 0 2px py of the system, and thus, the enpolarizing and depolarizing properties of the simulated samples should be conserved. Regarding data in Fig. 1(a)–(f), the first result we observe is that, here px and py are the amplitude attenuation coefficients for the x and y independently of the model parameters (px, py, and σθ), the P3 parameter components, respectively. To make the simulation more realistic, we always equals 1. Therefore, we cannot modify the P3 value with this will consider diattenuators oriented at different angles. To do so, we combination of diattenuators and the consequent depolarizers are al­ have to take into account the Mueller matrix of a rotator [28], where θ is ways restricted to the top of the tetrahedron of the Purity Space [35]. As the angle of the rotator. Therefore, the Mueller matrix of each linear a consequence of this, we cannot achieve an ideal depolarizer (diag(1,0, diattenuator rotated to an angle θ can be obtained as [28]: 0,0)) by simply varying the parameter values in the diattenuation ( ) M px , py , θ = Mrot ( − θ)MLD Mrot (θ). (6) model. Indeed, the P3 =1 plane does not contain the ideal depolarizer, and thus all the points on such a plane necessarily exhibit de­ Under this scenario, by applying the summation in Eq. (4) to Mueller polarizations strictly lower than the maximum one that is, depolariza­ matrices with the structure given in Eq. (6), we achieve a model with tion cannot be continuously increased to the ideal depolarizer value. different control parameters: the number n of Mueller matrices in the Secondly, we note that the values of P1 and P2 channels are modified summation, the amplitude coefficients (px and py) and the orientation with different values of px, py, and σθ, and thus, the depolarizing angle (θ) for each Mueller matrix. response of resulting depolarizers strongly depend on these parameters. Therefore, we need a criterion to choose the way to assign different As P3 is always equal to 1, this implies that different depolarizers values to the parameters in each diattenuator within the summation. (different pixels in images in Fig. 1(a)–(f)) correspond to different spatial These values could be set arbitrarily, for instance, by generating random positions at the plane P3=1 in the Purity Space [35]. Therefore, the numbers, but to simulate a situation closer to real samples, such as the channels P1 and P2 are suitable tools to differentiate between different distribution of collagen fibers in a tendon or cellulose in the vascular depolarizers originated by dichroic structures. Importantly, note the tube of a leaf, we restrict the domain of variation. We consider a group of close similarity in the responses of P1 and P2 images. In fact, they present dichroic optical elements, with a privileged direction, with deviations identical behavior for a fixed value of px (different rows), but P2 presents from this direction. To do that, the values for the control parameters in a vertical shift upwards as px increases (from bottom to upper row). This each of the n matrices in the summation are assigned by X + δ, where X situation can be explained by taking into account the inequality shown is the mean value parameter and δ is a random variable that follows a in Eq. (2), which forces P2 to have larger (or equal) values than P1. In Gaussian distribution with a null mean and a given variance (σ). The turn, we also observe that larger σ θ values (vertical axis) lead to smaller mean values for the amplitude coefficients px, py are bounded within the values for P1 and P2. In fact, for σ θ =0 we obtain P1=P2=P3=1 (yellow range (0, 1) and the mean orientation of the diattenuators within the color) in all the cases, and P1 and P2 tend to decrease their values as σ θ range (0,180º). increase. This situation implies that the larger the loss of organization Afterwards, for the sake of interpretation (to limit the space of sim­ for the constituent units (diattenuators) (i.e., σθ increase), the larger the ulations), we chose to fix in all the simulations the value of the ampli­ depolarizing capability of the system (i.e., P1 and P2 decrease). Note that tude coefficient in the x direction px to a constant value. Regarding the this behavior is observed until a certain limit value for σθ, that depends number of elements in the incoherent addition, we set n = 1500, this on the px value (different rows) but in general it occurs around σθ = 40º number being a trade-off between a large number of elements (to mimic both for P1 and P2 metrics. This implies that even increasing the disorder real processes of light-matter interaction) and a reasonable computation of polarizing units constituting samples, there is a certain limit from time. which depolarizing response of samples is not modified anymore. The Once we have set the values for px and n, our remaining control existence of this limit can also be discussed in an intuitive and visual way parameters are py and θ. For each value of px, we conduct a collection of in terms of the Purity Space, as presented in the Supplementary material simulations by taking a given value of px and changing the values of py 3 M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 Fig. 1. (a)–(f) IPP values for simulated samples composed by incoherent additions of linear diattenuators. The x axis represents the mean values of py and, the y axis the values for σ θ . The value of px is set as a constant for each image, with values of 0.8 and 0.2, respectively for the two different rows. The value of the mean orientation angle θ is set as 60º for all cases. (g) D value for the incoherent addition of linear diattenuators with σθ = 0, the x and y axis represent the px and py parameters, defined in the range 0 to 1. (Section 5). equal to 1 (or near) when the values of py are in the range (0,0.3). This This set of simulations can also be studied in terms of the diattenu­ corresponds with the left zone of the images where py is in the mentioned ation D observable, representing the dependence of the sample trans­ range and, we can see that in this case, for σ θ values higher than the mittance with the state of polarization of the incident light. The limit, the system depolarizes anisotropically. However, as py increases diattenuation D can be calculated as the absolute value of the diatten­ (D decreases) the depolarizing capability of the system is lost because uation vector, which can be written in terms of the first row Mueller the system becomes less dichroic. matrix elements in the following way [28]: Results shown in Fig. 1(g) corresponds with the case of σ θ =0. We have also tested other values for the standard deviation of the mean 1 D= (m01 , m02 , m03 )T . (7) orientation of the diattenuators and results and discussion are provided m00 in the supplementary Material. The main conclusions are that the gen­ If we substitute in Eq. (7) the elements of the Mueller matrix corre­ eral structure shown in Fig. 1(g) is maintained, but the diattenuation spondent to a general diattenuator (Eq. (5)), the absolute value of the values decrease with σθ, reaching a limit scenario, where diattenuation diattenuation vector, D, can be expressed as [17]: is equal to 0 for σθ ≥40º. This result has sense because as larger the ⃒ ⃒ 2 ⃒ ⃒ deviation of θ, as larger the depolarizing capability of the depolarizer ⃒px − p2y ⃒ consisting of dichroic unitary elements. D= 2 . (8) px + p2y Summarizing, in this subsection we have mimicked multiple de­ polarizers consisting of dichroic microscopic elements. As expected, Note that the diattenuation parameter D is a significant feature of incoherent addition of isotropic (in terms of dichroism) units (i.e., px=py diattenuators, taking values of D = 1 for an ideal diattenuator, and scenario), lead to non-depolarizing samples (i.e., P1=P2=P3=1). In turn, values of D = 0 for a medium with no diattenuation. in the case of anisotropic elements (px∕ = py), the resulting systems Taking the definition of D in mind, and back again to Fig. 1(a)–(f), we become depolarizers, and their depolarization capability increases for find a non-depolarizing zone (yellow color) when px=py (indicated with larger diattenuation D values (which depends on the px and py relation) a red dotted line in each image). According to Fig. 1(g) this situation of the constituent diattenuators, as well as with the units orientation corresponds to values of D = 0 (see positive diagonal in blue), and disorder σθ. Importantly, the value of P3 is independent of all the model therefore, non polarimetric elements (thus without the capability of parameters, and it maintains a constant value of P3=1 for all the simu­ enpolarize and depolarize). This condition leads to simulations based on lations, and therefore, all depolarizers consisting of collections of linear the incoherent addition of media represented by the identity matrix, and diattenuators are placed into a specific plane of the Purity Space [35]. as a consequence, without the potential of implementing depolarizers (see red dotted line in Fig. 1(a)–(f)). However, for px∕ = py, the units of the incoherent addition present certain diattenuation (i.e., D∕ =0) (see Fig. 1 3.2. Depolarizers originated by the incoherent addition of linear retarders (g)), which leads to systems with certain depolarizing response. This depolarization response directly depends on the organization disorder The second study considers depolarizers consisting of microscopic (σ θ) (see vertical axis in Fig. 1(a)–(g)). unit elements presenting linear retardance. We proceed in analogy to the The dependence of the IPP with the px and py values is easily un­ previous study corresponding to diattenuation, building in this case derstood by studying Fig. 1(g). The pink and the white dashed lines in models consisting in the incoherent sum of sets of retarders as a function the figure, indicate the D value for px=0.8 and px=0.2, respectively, with of their retardation and orientation. In Ref. [36] it is shown how the py taking values from 0 to 1. Following the dashed white line in Fig. 1(p), incoherent addition of linear retarders can also be performed analyti­ the value of D is maximum for py=0 and start to decrease as py increases, cally. We show the Mueller matrix for a linear retarder oriented at 0º reaching the minimum for py=0.2 (i.e., px=py). As py increases, the value [28]: of D increases as well, reaching the D = 1 value. This behavior explains ⎛ 1 0 0 0 ⎞ the depolarization results obtained from Fig. 1(d) and (e), where ⎜0 1 0 0 ⎟ P1=P2=1 (non depolarizing scenarios) correspond to values of D equal MLR = ⎜ ⎟ ⎝ 0 0 cosϕ sinϕ ⎠; 0 ≤ ϕ ≤ π , (9) or very close to zero (i.e., corresponding to the blue diagonal in Fig. 1 0 0 − sinϕ cosϕ (g)), whereas for the situations where D reaches its maximum of high values, the depolarization capability of the samples increase, achieving where, ϕ is the phase shift introduced to the orthogonal components of values of P2 and P1 different from 1. The pink line indicates the case for the incident light field. As in the previous case, to make simulations px= 0.8. If we study the D behavior through this line, we can see that D is more realistic, we consider an ensemble of retarders oriented at different 4 M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 angles, according to the relation M = Mrot ( − θ)MLR Mrot (θ). For the (40º–320º). Within this scenario, the orientation disorder of the re­ incoherent sum of Mueller matrices of retarders, we consider a mean tarders units (given by the parameter σθ) becomes significant, and de­ orientation (θ) for the retarders with a standard deviation (σ θ) following polarization is observed for σθ values larger than 20º: P1 taking values a probabilistic Gaussian distribution. In addition, we vary the retardance from 0.9 to 0 and P2 from 1 to 0. value ϕ from 0º to 360º and with a null variance. As in the previous case, In analogy to diattenuators, it seems that there is a limit value of σθ, the obtained simulations (which are functions of ϕ and σθ) are inter­ above which depolarization does not increase. For the case of retarders, preted in terms of the P1, P2, P3 metrics. such limit value is around 20º (see more details in Section 5 of the In Fig. 2 we present the three IPP metrics corresponding to the supplementary text). simulations with linear retarders. The x axis represents the mean values Note that, for values of ϕ close to 180º, P1 becomes 0 and P2 is equal of retardance ϕ, taking values from 0º to 360º, whereas the y axis rep­ to 1. In this case, the Eq. (9) becomes diag(1,1,− 1,− 1), and therefore, resents the variance range from the mean orientation value, σθ, which only P1 is able to differentiate between this case and the above-discussed can take values from 0º to 180º. For each of the possible values of ϕ we identity matrix scenario (ϕ=0º or 360º). Accordingly, this difference build the M corresponding to the incoherent sum of n = 1500 Mueller between P1 and P2 can be useful to distinguish between different de­ matrices where each of them has a value of orientation with a variance polarizers consisting of linear retarders with different retardance ϕ given by a Gaussian distribution. This process is repeated for the possible values. For example, a retardance value of ϕ=90º leads to P1 and P2 values of ϕ and σθ, both can take 300 different values within their values close to zero, a value of ϕ =180º to P1=0 and P2=1 and a value of respective ranges, this leads to 300×300 (90,000) Mueller matrices ϕ=0º or 360º to P1=1 and P2=1. representing different depolarizers based on retarders. Afterwards, we Finally, as a complementary material, in the supplementary text calculate the IPP values of these matrices according to the calculus given (Fig. S3) we provide the results for further simulated scenarios for the Section 2. As in the previous case, depolarization response of the linear retarders case, for instance by adding a variance σ ϕ (based on a simulated systems is independent of the mean orientation of the re­ Gaussian probability) to the mean retardance ϕ. Note that even though tarders, as system rotations do not represent changes in its physical different simulations lead to different values for P1 and P2 channels, in properties. Therefore, the mean orientation of the unit retarders is all the cases, the value of P3 remains always constant to P3=1. arbitrarily set to θ= 60º in conducted simulations. Summarizing, the simulations shown in this subsection further re­ In Fig. 2 the values of P1 and P2 vary from 0 to 1 depending on the inforces the idea that depolarizing systems originated by diattenuators combination of (σθ, ϕ), whereas P3 is always 1. Therefore, as it happened or linear retarders microscopic units, are always placed at the top plane in the case of systems consisting of dichroic elements, systems based on (P3=1 plane) of the tetrahedron of the Purity space. We also observe that linear retarders are not able to decrease the value for P3. In terms of P1 and P2 parameters strongly depend on the model parameters: atten­ Purity Space, different depolarizers consisting of linear retarders are uations (px, py), retardance (ϕ) and the variance of the orientation (σθ), distributed as well within the plane P3=1. As already discussed in the as well as the diattenuation D. These two metrics can be useful to reveal diattenuation section, due to this restriction of P3=1 in the retardance underlying properties of the systems, as if their main constituents are case we cannot achieve an ideal depolarizer (diag(1,0,0,0)) by simply based on diattenuators, linear retarders, or a mix of them. Therefore, varying the parameter values in the retardance model, as it is not con­ they can be used to discriminate between depolarizers based on different tained in such a plane. linear retarders configurations, or even to differentiate between de­ Unlike the previous case, where P1 and P2 showed a very similar polarizers based on dichroic or on linear retarder constituent elements. behavior (just modified by a vertical shift), in the case of linear retarder Under this scenario, a question arises. Which kind of structures are able based systems, the P1 shows one minimum valley instead of the two to decrease the P3 parameter? This situation is explored in the following shown by the P2 channel (i.e. the P1 distribution between ϕ [0–360º] section. range occurs as well for the P2 channel, but in the ϕ [0–180º] range, and it is doubled in the ϕ [0–360º] range). Obviously, according to inequality 4. Depolarizing systems with P3<1 shown in Eq. (2), the vertical shift in P2 with respect P1 is also present. In detail, for retardances ϕ lower than 40º and higher than 320º the In practical situations, it is common to find depolarizing systems simulated samples become non depolarizing (yellow color) indepen­ (both organic and inorganic) with values of P3 smaller than one, even dently of the σ θ value, having P1=P2=P3=1. This is because for values of achieving very low values (close to zero). For instance, in plant and retardance ϕ close to 0º and 360º, Eq. (9) becomes the identity, and animal tissue we can find P3 values ranging from 0 to 1 in different therefore, the resulting systems loss their depolarizing capability. structures [24,37]. Note that some of these samples indeed consist of Interestingly, as we go far from such values, retardance behavior be­ anisotropic elementary components (for instance, animal tissues are comes significant and the resulting systems become depolarizers (going mainly based on collagen fibers [38], which can be described by re­ gradually from yellow to blue in Fig. 2(a) and (b)). In particular, tarders, or vegetal samples are rich in dichroic units [39,40], which can depolarizing behavior of systems is achieved for values of ϕ in the range be described by diattenuators). Fig. 2. IPP values for simulated samples composed by linear retarders; (a) P1, (b) P2 and, (c) P3, where x axis represents the mean values of ϕ simulated and, the y axis the variance range values from the from the mean orientation θ= 60º. 5 M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 Recalling the results of previous Section 3, we realize that aniso­ addition, as discussed before, the depolarization introduced by the tropic constituent elements in samples cannot explain why all those real ̂ iso term cannot relate to systems based on anisotropic constituent βm00 M samples led to values of P3<1, and therefore, it may exist another elements (retarders, diattenuators, mix of them etc.) and thus, it must be depolarizing mechanism to be considered. connected with isotropic processes introducing polarimetric random­ Under this scenario, this section focuses on finding a physical ness, as it is the case of isotropic scattering. For this reason, from now on, mechanism able to describe the depolarizing behavior of real samples let us call the first term of Eq. (12) as the anisotropic depolarization with P3<1. We want to note that there are different mechanisms leading term, and the second term as the isotropic depolarization term. to P3<1, but we only focus on those that seem more adequate to explain The relation between P3 and the new added term in the simulation is the depolarizing behavior of most real samples. Some uncommon exotic studied by a set of simulations. By following such Eq. (12), as in previous mechanisms leading to P3<1 (some of them even not representing Section 3, we have access to different control parameters (orientations, physical systems) are provided in Section 2.1 of the supplementary retardances, absorption coefficients, deviations, etc.) involved in the document. anisotropic term, but now, we also control the isotropic component In this section we introduce a new group of simulations that lead to relevance through the weight of the parameter β. We conducted a full P3<1. As in previous Section 3, they consist of an incoherent addition of space of simulations by repeating the cases analyzed in previous Section a number of M representing basic systems (linear retarders and dia­ 3, but now, by adding the effect of the isotropic term. For the sake of ttenuators), but now, we add an extra term in the summation, this being simplicity, in the following we only show and discuss simulations cor­ the Mueller matrix of a perfect depolarizer with a certain weight (β). responding to a particular case, the incoherent addition of linear dia­ The addition of this term in the simulations is inspired on the char­ ttenuators (Section 3.1) but generalized by the isotropic term. Further acteristic decomposition. As can be seen in Eq. (3), this decomposition simulations results (retarders, combination of linear diattenuators and describes a general matrix M as the addition of four terms, where each retarders, etc.) generalized with the isotropic term, can be found in the one of them is weighted by a linear combination of the IPP. The first supplementary text (see Fig. S6), but main conclusions are analogous for term carries the fully polarized contribution of the system. The second all these cases. and third terms, represent the depolarizing contribution of 2D and 3D Simulated results corresponding to the case of linear diattenuator systems [17]. Finally, the last term depends on a perfect depolarizing systems (anisotropic term) incoherently added with the isotropic term medium (diag(1,0,0,0)), which completely depolarizes any incident are shown in Fig. 3. Results are given in terms of the P3 metric, P1 and P2 polarization state of light independently of its initial polarization state. have the same dependence with the anisotropic term parameters as in Interestingly, the IPP corresponding to a perfect depolarizer are the previous section. As the value of β increases (i.e. P3 decreases), the P1=P2=P3=0, and thus, such systems have the capability of decreasing value of P1 and P2 also decrease (see Eq. (2)) but maintaining the the P3 channel [41]. dependence with the anisotropic parameters. To implement the corre­ For these reasons, we realize that the addition of a perfect depolar­ sponding simulations, the control parameters associated to the aniso­ izer contribution to the simulations described in previous Section 3 may tropic term were set as: absorption coefficient in the x direction, px, set lead to a decrease of the P3 value. Therefore, in the new set of simula­ to 1, mean orientation of linear diattenuators, θ, set to 60º, absorption tions, we add an additional term representing a perfect depolarizing coefficient in the y direction, py, taking values from 0 to 1 (values of x Mueller matrix with a certain weight that can be controlled. In partic­ axis) and, orientation deviation from mean value, σθ, taking values from ular, this new system can be written as the incoherent addition of a first 0º to 180º (values of y axis). In addition, from Fig. 3(a) to (e) the weight term describing the contribution to depolarization originated by aniso­ of the isotropic term, β (see Eq.(12)), increases in steps of 0.2 from β tropic (linear retarders, diattenuators) constituent elements (and ob­ =0 to β =1. tained as described in Eq. (4)), with a second term, Miso, representing the In Fig. 3 we see that, as expected, the P3 metric does not depend on perfect depolarizing or isotropic contribution. The resulting M can be the control parameters of the anisotropic term, i.e., all the images in written as follows: Fig. 3 show a homogeneous color (i.e., present the same value for P3), n′ because P3 values are independent of x and y axes values. This situation ∑ ∑ m ̂ = M = m00 M ̂ i) + αi (m00 M ̂ iso ); βj (m00 M occurs independently of the anisotropic term control parameters chosen for x and y axes and the elements conforming this term (see Sections 2.2 (10) i j n′ ∑ ∑ m and 6 of the supplementary material). Unlike this, the values of the αi , βj ≥ 0; αi + βj = 1, isotropic weight, β, control the values of the P3 Purity parameter (see i=1 j=1 Fig. 3(a)–(c)). In particular, as larger the β value as shorter the P3 (note the colorbar in Fig. 3). For the sake of clarity, we also provide Fig. 3(d) where, n’+m = n. Note that as in Section 3, n gives the total number of where we directly represent the relation between β and P3, where we terms in the summation (we set n = 1500 once again). Since the Misois note an inverse linear relation between these two parameters. We want the Mueller matrix of a perfect depolarizer (diag(1,0,0,0)) the second to note that the values for the px and σθ selected for the simulations are term can be written as: arbitrary and we could set other values within the range of the param­ ∑ m eters: px [0,1] and σθ [0,180]. Nevertheless, the main conclusions would ̂ iso ) = βm00 M βj (m00 M ̂ iso , (11) remain the same. According to Eq. (12), results in Fig. 3(d) have been j obtained from a Mueller matrix composed by an anisotropic part where, β = ∑m βj . Therefore, the Eq. (10) can be expressed as: (incoherent addition of diattenuators) and the isotropic part (diag j (1,0,0,0)). To cover all the possible range of P3 values, we conduct n′ ∑ simulations with β from 0 to 1. This situation confirms that the amount ̂ = M = m00 M ̂ i ) + βm00 M αi (m00 M ̂ iso . (12) of isotropic scattering in samples is described by the P3 metric and is directly related to the weight of the isotropic term in Eq. (12). Moreover, i ∑′ The first term, ni αi (m00 M ̂ i ), represents the effects of depolarizing since M ̂3 = M ̂ iso , by comparing Eq. (12) with the characteristic systems already described in Section 3, that is, depolarizing systems decomposition (Eq. (3)) we can find the following relation: originated by component elements showing retardance or dichroism, ̂ iso = (1 − P3 )(m00 M βm00 M ̂ 3) (13) and leading to P3=1. In turn, the new added term, βm00 M ̂ iso , corresponds to depolarizing systems that fully depolarize light, independently of the and thus, β = 1- P3 (note that this result is in agreement with previous input state of polarization of the illumination, and leading to P3=0. In Fig. 3(d)). In this way, we find the relation between the weight of the 6 M. Canabal-Carbia et al. Optics and Lasers in Engineering 172 (2024) 107830 Fig. 3. P3 values for the simulated samples composed by the incoherent addition of linear diattenuators and a pure depolarizing component with different weights: β=0.4 (a), β=0.6 (b) and β=0.8 (c). The x axis represents the mean value of py (where px=1) and the y axis the variance range values of the orientation (σθ) from 0º to 180º. (d) Relation between β and P3 obtained from a Mueller matrix with isotropic and anisotropic component, the weight of each component is given by the β value and the parameters that control the anisotropic term are set as px=1, σθ=45º and py varies in the range [0,1]. isotropic contribution, β, with P3. Moreover, considering the relation related to constituent elements showing retardance and/or dichroism between αi and β weights in Eq. (10), together with β= 1- P3, we obtain: and anisotropic scattering produced by these elements. Therefore, the larger the value of P3, the more relevant become P1 and P2 metrics. In n′ n′ n′ 1= ∑ αi + β = ∑ αi + (1 − P3 )→ ∑ α i = P3 . (14) this vein, is interesting to note the particular case where P3=P2=P1. In i=1 i=1 i=1 this case, if we study the terms in the characteristic decomposition Eq. (3) the two terms corresponding to the M representing the anisotropic By taking into account these results, we can reformulate the weights information of the sample disappear. Therefore, this particular case of the two terms in Eq. (12) just as a function of P3. To this aim, we define represents a sample where the depolarization is due only to isotropic a normalized Mueller matrix M ̂ A in the following way: processes and the parts of the sample representing MA only contribute to n′ n′ the non-depolarizing term. ∑ 1 ∑ ̂ A →M αi Mi = P3 M ̂A = αi Mi . (15) Summarizing, in this section we have demonstrated that the depo­ i P3 i larizing response of samples can be categorized in two subgroups: (1) This relation allows us to re-write Eq. (12) as: those originated by isotropic processes; and (2) those originated by anisotropic processes. These two origins can be produced simulta­ ̂ A + (1 − P3 )(m00 M M = P3 m00 M ̂ 3 ), (16) neously in samples, but their significance can be studied in terms of the Indices of Polarimetric Purity (P1, P2 and P3 metrics). On the one hand, where, M ̂ A is defined by Eq. (15) and M ̂3 = M ̂ iso . Note that considering the P3 metric gives a measure of the presence of depolarization behavior the equivalence of Eqs. (16) and (12), the first term, m00 M ̂ A , is the originated by isotropic processes. On the other hand, by considering results provided in Section 3, the metrics P1 and P2 complete this in­ anisotropic term; and the second term, m00 M ̂ 3 , is the isotropic term, and formation by describing characteristics of anisotropic origin of depo­ their significance in the incoherent addition is controlled by the Purity larization (as dichroism, retardance, or a mixture of them). Moreover, term P3. the difference between the IPP (P2-P1 and P3-P2) gives information about Under this scenario, we can say that P3 controls the depolarizing the amount of anisotropic depolarization: more difference between the origin of samples. When P3=1 the isotropic term is zero (no isotropic IPP implies more anisotropic depolarization, representing only isotropic depolarization is present), and thus, all the depolarization behavior of depolarization processes when such difference is zero (i.e., P1=P2=P3 samples is given by the anisotropic term, hence, depolarization is orig­ case). More insight in this topic can be obtained by studying how a inated by the intrinsic polarimetric anisotropies of the sample. In particular depolarizer depolarizes the fully polarized states of polari­ contrast, for P3=0, the anisotropic term is cancelled and polarimetric zation at the surface of the Poincaré Sphere and study this situation in anisotropies of constituent elements do not contribute to the depola­ terms of IPP [17,42]. A brief discussion in this regard is also provided in rizing behavior. In such a case, all depolarization is due to the isotropic Supplementary material. term, and corresponding samples behave as perfect depolarizers, origi­ Note that this is a fundamental result providing which phenomenon nated by processes such as isotropic scattering. In the regime 0