Polarization-Resolved Digital Holographic Sensing for Simultaneous Morphological and Polarimetric Characterization of Microparticles

1. Introduction

Micrometer-sized particles are ubiquitous in atmospheric environments, biopharmaceutical formulations, and industrial processes. Their size, morphology, spatial distribution, and optical properties are often required simultaneously because these quantities describe complementary aspects of particle state and composition. In atmospheric research, particle concentration, size distribution, and morphology are important for pollution monitoring, exposure assessment, and cloud microphysical studies [1,2,3]. In biopharmaceutical applications, the number, size, morphology, and optical characteristics of protein aggregates, silicone-oil droplets, and glass fragments are relevant to product quality and stability [4]. These demands are driving particle sensing from single-parameter measurements toward multiparameter characterization at the individual-particle level [5,6].

Existing particle-measurement techniques provide different subsets of this information. Optical particle counters and single-particle light-scattering instruments rapidly estimate particle size and number concentration [7,8], while optical array probes, microscopic imaging, and micro-flow imaging provide two-dimensional size and shape information [3,9,10]. Electron microscopy, Raman spectroscopy, and Fourier-transform infrared spectroscopy can provide finer morphological or material-composition information, but generally require different measurement configurations and do not inherently provide three-dimensional particle localization [4,6]. Polarimetric measurements offer a complementary optical dimension because the polarization response of scattered light depends on refractive index, shape, internal structure, orientation, and scattering geometry [11,12]. Ensemble polarimetry and lidar have demonstrated the value of polarization for characterizing suspended particles and aerosols [13,14,15], whereas single-particle polarimetry can retain particle-to-particle heterogeneity. For example, microfluidic or flow-based systems have been used to guide particles through a defined polarization-scattering region for individual-particle analysis [16,17]. However, these approaches typically rely on constrained flow channels, fixed scattering volumes, or limited focal depths.

A key sensing challenge therefore arises for micrometer-sized particles distributed over a three-dimensional measurement volume. Conventional imaging has a finite depth of field: particles away from the focal plane are blurred, and their signals can overlap with neighboring particles, making particle-resolved morphological and polarimetric measurements difficult. Digital holography addresses the axial-localization problem by recording a hologram from which particles at different depths can be numerically refocused and localized in three dimensions [18,19,20]. Polarization-sensitive digital holography has also been used for quantitative polarization imaging, birefringence measurement, and material characterization [21,22,23,24,25,26,27]. In particular, previous studies have demonstrated simultaneous holographic and polarization recording for microfibers and microplastics [25,26,27]. These studies establish the complementarity of holography and polarimetry, while leaving room for a sensing framework that associates three-dimensional particle localization and morphology with particle-resolved linear-polarization parameters within the same particle field.

In this work, we develop a polarization-resolved digital holographic sensing system for simultaneous morphological and polarimetric characterization of individual microparticles within a three-dimensional particle field. The system combines digital in-line holographic reconstruction with division-of-focal-plane polarization imaging to retrieve, for each resolved particle, three-dimensional coordinates, equivalent diameter, major and minor axes, perimeter, area, and shape factor together with S₀, S₁, S₂, and DoLP. The particle-sizing capability is evaluated using three groups of standard glass microspheres. We then examine how the measured DoLP varies across particle groups with different nominal refractive indices and sizes and use T-matrix calculations to test whether the experimentally observed trend is physically consistent with forward-scattering polarimetry. The main contributions are therefore twofold: (1) particle-resolved, co-registered retrieval of morphological and linear-polarization parameters within a three-dimensional particle field; and (2) experimental identification and numerical interpretation of a refractive-index-dependent DoLP response under the investigated measurement conditions.

2. Measurement Principles

2.1. Polarization Measurement and Refractive-Index Sensitivity

For an interface between two dielectric media, the Fresnel coefficients for s- and p-polarized light depend on the refractive indices and incidence geometry [28]:

$$r_s={\displaystyle \frac{n_1\cos {\theta }_i-n_2\cos {\theta }_t}{n_1\cos {\theta }_i+n_2\cos {\theta }_t}}$$

(1)

$$r_p={\displaystyle \frac{n_2\cos {\theta }_i-n_1\cos {\theta }_t}{n_2\cos {\theta }_i+n_1\cos {\theta }_t}}$$

(2)

where n₁ and n₂ are the refractive indices of the media, θ_i is the angle of incidence, and θ_t is the angle of refraction, satisfying Snell’s law:

$$n_1\sin {\theta }_i=n_2\sin {\theta }_t$$

(3)

$$\cos {\theta }_t=\sqrt{1-{\left({\displaystyle \frac{n_1}{n_2}}\right)}^2{\left(\sin {\theta }_i\right)}^2}$$

(4)

The Fresnel relations illustrate, at the interface level, why refractive-index contrast can modify the relative amplitudes of orthogonal polarization components. For an incident field resolved into p- and s-polarized components, the reflected components can be written as:

$${\mathit{E}}_{\mathit{r}\mathit{e}\mathit{f}}=\left[\begin{array}{c}{\mathit{E}}_{\mathit{p}}^{\mathit{r}\mathit{e}\mathit{f}}\\ {\mathit{E}}_{\mathit{s}}^{\mathit{r}\mathit{e}\mathit{f}}\end{array}\right]=\left[\begin{array}{c}{r}_p{\mathit{E}}_{\mathit{p}}^{\mathit{i}\mathit{n}}\\ r_s{\mathit{E}}_{\mathit{s}}^{\mathit{i}\mathit{n}}\end{array}\right]$$

(5)

Here, E_p and E_s denote the p- and s-polarized electric-field components, and r_p and r_s are the corresponding Fresnel reflection coefficients. These relations are introduced only to illustrate the refractive-index sensitivity of polarization at an optical interface; the PRDH measurements in this study are not interpreted as a direct Fresnel-reflection inversion.

The polarization state of a partially polarized optical field can be represented by the four Stokes parameters:

$$\begin{array}{c}{\mathit{S}}_0=〈{\left|{\mathit{E}}_{\mathit{p}}\right|}^2〉+〈{\left|{\mathit{E}}_{\mathit{s}}\right|}^2〉=A_p^2+A_s^2\\ {\mathit{S}}_1=〈{\left|{\mathit{E}}_{\mathit{p}}\right|}^2〉-〈{\left|{\mathit{E}}_{\mathit{s}}\right|}^2〉=A_p^2-A_s^2\\ {\mathit{S}}_2=2R_m〈{\mathit{E}}_{\mathit{p}}{\mathit{E}}_{\mathit{s}}^{\mathit{*}}〉=2A_p{A}_s\cos \left({\varphi }_p-{\varphi }_s\right)\\ {\mathit{S}}_3=2I_m〈{\mathit{E}}_{\mathit{p}}{\mathit{E}}_{\mathit{s}}^{\mathit{*}}〉=2A_p{A}_s\sin \left({\varphi }_p-{\varphi }_s\right)\end{array}$$

(6)

Here, E_p and E_s are the orthogonal electric-field components; R_m and I_m denote the real and imaginary parts, respectively; E_s^* is the complex conjugate of E_s; A_p and A_s are the corresponding amplitudes; and ϕ_p and ϕ_s are their phases.

For intensity-based polarization detection, the Stokes parameters can be expressed in terms of analyzer-resolved intensities [29]:

$$\mathit{S}=\left(\begin{array}{c}{\mathit{S}}_0\\ {\mathit{S}}_1\\ {\mathit{S}}_2\\ {\mathit{S}}_3\end{array}\right)=\left(\begin{array}{c}{I}_{0^{°}}+I_{{90}^{°}}\\ I_{0^{°}}-I_{{90}^{°}}\\ I_{{45}^{°}}-I_{{135}^{°}}\\ I_R-I_L\end{array}\right)$$

(7)

The degree of polarization (DoP) is defined as the polarized fraction of the total optical intensity:

$$DoP={\displaystyle \frac{\sqrt{{\mathit{S}}_1^2+{\mathit{S}}_2^2+{\mathit{S}}_3^2}}{{\mathit{S}}_0}}$$

(8)

When only the linear polarization components are considered, the degree of linear polarization (DoLP) is calculated as [30]:

$$DoLP={\displaystyle \frac{\sqrt{{\mathit{S}}_1^2+{\mathit{S}}_2^2}}{{\mathit{S}}_0}}$$

(9)

DoLP ranges from 0 to 1 and quantifies the fraction of the measured intensity associated with linear polarization. In the present PRDH system, S₀, S₁, and S₂ are obtained from the four analyzer orientations of the DoFP camera, and DoLP is then calculated from Equation (9). Because the recorded hologram contains the unscattered transmitted field together with the particle-scattered field, the measured DoLP is influenced jointly by particle refractive index, size, shape, and the collection geometry. Accordingly, the refractive-index dependence reported in this study is evaluated empirically and interpreted with T-matrix scattering calculations rather than treated as a direct refractive-index inversion.

The interface-level relations above therefore provide qualitative physical motivation, whereas the quantitative particle response is governed by electromagnetic scattering. Section 5 uses the T-matrix method to examine the dependence of the forward-scattering polarization response on particle size and refractive index.

2.2. Digital In-Line Holographic Reconstruction

Digital in-line holography (DIH) provides a compact route to volumetric particle sensing because a single recorded hologram can be numerically propagated to multiple axial planes. When a plane wave illuminates a particle, the particle-scattered field interferes with the unscattered transmitted field, and the resulting interference pattern is recorded as a hologram. The object field can then be numerically reconstructed using scalar diffraction propagation. For a measurement volume containing multiple particles, the reconstructed field at an axial distance Z_r can be written as:

$${\mathrm{U}}_{\mathrm{R}}\left(\mathrm{u},\mathrm{v}\right)={\displaystyle \frac{1}{\mathrm{j}\mathsf{\lambda }}}\underset{\mathrm{\infty }}{\iint }\mathit{R}\left(\mathit{x},\mathit{y}\right){\mathrm{I}}_{\mathrm{H}}\left(\mathrm{x},\mathrm{y}\right){\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{j}\mathrm{k}\sqrt{{\left(\mathrm{u}-\mathrm{x}\right)}^2+{\left(\mathrm{v}-\mathrm{y}\right)}^2+{\mathrm{Z}}_{\mathrm{r}}^2}\right)}{\sqrt{{\left(\mathrm{u}-\mathrm{x}\right)}^2+{\left(\mathrm{v}-\mathrm{y}\right)}^2+{\mathrm{Z}}_{\mathrm{r}}^2}}}\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{y}$$

(10)

where λ is the wavelength; R(x,y) denotes the reference field; I_H(x,y) is the recorded hologram intensity; k is the wave number; x and y are coordinates on the hologram plane; u and v are coordinates on the reconstruction plane; and Z_r is the reconstruction distance. By propagating the hologram through a sequence of reconstruction distances, particles located at different axial positions can be brought into focus and localized in three dimensions [31].

After the particle focal plane is determined, contour-based image analysis is used to extract particle morphology. A minimum-area bounding rectangle is fitted to the segmented particle contour using a rotating-calipers procedure [32,33], providing the major and minor axes. The particle area S_i , major-axis L and contour perimeter L_i are obtained from the binary particle region, and the equivalent diameter is calculated from the segmented area. The shape factor F is calculated using Equation (11); for an ideal circular projection, F approaches 4, and larger values indicate greater deviation from a compact circular shape [34,35].

$$\mathrm{F}={\displaystyle \frac{\mathrm{L}\times {\mathrm{L}}_{\mathrm{i}}}{{\mathrm{S}}_{\mathrm{i}}}}$$

(11)

3. Materials and Methods

3.1. PRDH Sensing System

The PRDH sensing system integrates digital in-line holography with division-of-focal-plane polarimetry so that holographic and linear-polarization information are recorded under the same illumination condition. This architecture is designed to support particle-resolved retrieval of morphology and polarization within a three-dimensional measurement volume. The optical layout is shown in Figure 1a.

A 520 nm pulsed laser is expanded by L1 and collimated by L2 to provide approximately uniform illumination. The beam then passes through the linear polarizer L3 (extinction ratio > 1000:1). Beam splitter L4 divides the polarized beam into a monitoring arm and a measurement arm. The monitoring arm is recorded by DoFP1 to monitor the incident polarization state, while the measurement arm illuminates the particle field. The particle-scattered field and the unscattered transmitted reference field are collected by a 5× telecentric lens (WWK50-102-111V3; 102 mm working distance; distortion 0.16%; telecentricity 0.18°) and recorded by DoFP2. DoFP2 uses a Sony IMX264 global-shutter sensor with a 5-megapixel resolution and simultaneously records analyzer-resolved intensities at 0°, 45°, 90°, and 135° at up to 36 frames/s at full resolution. The simultaneous four-orientation acquisition provides the intensity information required for linear Stokes and DoLP retrieval while the holographic interference pattern provides the information required for numerical refocusing and three-dimensional localization [36].

3.2. Particle Samples and Evaluation Design

The PRDH system was evaluated using droplets, borosilicate glass microspheres of different nominal sizes, soda-lime glass microspheres, and irregular sand grains. The experiments were organized to examine three complementary aspects of performance. First, standard glass microspheres were used to evaluate particle-size measurement. Second, 5 and 10 μm borosilicate microspheres were compared to examine the effect of size at the same nominal refractive index. Third, particle groups spanning nominal refractive indices from approximately 1.33 to 1.52 were used to examine the dependence of the measured DoLP on refractive index under the present optical configuration. Because the particle groups differ in more than one physical property, the refractive-index analysis is interpreted as an empirical dependence rather than as a direct refractive-index calibration.

4. Results

4.1. Morphological Parameter Retrieval and Particle-Size Validation

Figure 1d illustrates numerical reconstruction of a polarization-resolved hologram containing droplets. The recorded hologram was propagated through 35 axial reconstruction planes separated by 5 mm over a total range of 170 mm. This axial stack captures the evolution of particle focus with reconstruction distance and provides the basis for determining the particle focal plane.

The transfer function employed by the convolution method is:

$$H\left(z\right)=\exp \left(i2\pi z\sqrt{\left({\displaystyle \frac{1}{{\lambda }^2}}-{\left({\displaystyle \frac{x}{X\delta }}\right)}^2-{\left({\displaystyle \frac{y}{Y\delta }}\right)}^2\right)}\right)$$

(12)

where λ = 520 nm is the laser wavelength, δ = 3.45 μm is the sensor pixel size, x/Xδ is the normalized spatial coordinate in the horizontal direction, and y/Yδ is the normalized spatial coordinate in the vertical direction.

Particle focus was determined using the local intensity-gradient variance. For each candidate particle region, the gradient variance was calculated across the 35 reconstructed planes, and the plane with the maximum value was selected as the in-focus plane. The corresponding reconstruction distance provides the axial coordinate, while the particle centroid in the selected plane provides the transverse coordinates. An example of an in-focus reconstruction is shown in Figure 1e.

Morphological parameters were then extracted from the in-focus particle image. Figure 2 shows representative processing results for (a) a 10 μm borosilicate microsphere, (b) a 50 μm soda-lime glass microsphere, and (c) a sand grain. The reconstructed in-focus images are shown in Figure 2((a-3),(b-3),(c-3)).

Each in-focus particle was segmented by intensity thresholding followed by morphological opening and closing to suppress isolated pixels and smooth the particle boundary. When enclosed cavities remained in the segmented region, hole filling was applied to obtain a continuous particle mask. The resulting masks are shown in Figure 2((a-4),(b-4),(c-4)).

For each segmented particle, the projected area and contour perimeter are calculated from the particle mask. A minimum-area bounding rectangle is fitted to the extracted contour to obtain the major and minor axes, and the equivalent diameter is calculated from the particle area. Together with the centroid coordinates and reconstruction distance, the procedure yields nine scalar morphological descriptors for each particle: x, y, and z coordinates; equivalent diameter; major axis; minor axis; perimeter; area; and shape factor F. The shape factor is calculated using Equation (11). Table 1 lists these quantities for the three representative particles in Figure 2.

To evaluate particle-size measurement, three groups of standard glass microspheres with nominal diameters of 5.1 ± 0.5 μm, 10.0 ± 1.0 μm, and 49.0 ± 1.4 μm were measured. For convenience, these groups are referred to as the 5, 10, and 50 μm particles. As shown in Figure 3, the measured values are distributed around the corresponding nominal sizes. The average absolute relative deviation between the group-mean measured diameters and the nominal diameters is approximately 5.5%. This result provides a quantitative check of the sizing performance under the present reconstruction and segmentation settings. Residual deviations can arise from the discrete axial reconstruction interval and from threshold-dependent localization of the particle boundary; therefore, the reported value should be interpreted as a system-level sizing result for the present measurement conditions rather than as a universal accuracy specification.

Figure 4 compares the shape factor F for particle groups with different morphologies. Droplets and irregular sand grains exhibit clearly different distributions because their projected shapes differ substantially, whereas the distributions of droplets and similarly sized borosilicate microspheres overlap strongly because both groups are approximately spherical in projection. These results illustrate that morphology provides useful but incomplete particle information and motivate the addition of a co-registered optical parameter that is sensitive to material-dependent scattering.

Accordingly, morphological descriptors and polarimetric descriptors are treated as complementary measurements in the PRDH framework. The following section examines the linear-polarization response of the same particle groups, with particular attention to its dependence on nominal refractive index and particle size.

4.2. Polarimetric Characterization

For each particle, the analyzer-resolved holograms are reconstructed at the particle-specific focal plane determined above. The corresponding intensity values are then used to calculate S₀, S₁, and S₂ according to Equation (7), and DoLP is calculated using Equation (9). The particle mask obtained from the morphological channel is used to associate the polarization measurement with the same spatially resolved particle. Figure 5 shows representative DoLP maps, where warmer colors denote higher DoLP and cooler colors denote lower DoLP.

Figure 6a–e summarize the particle-averaged DoLP values for the five investigated groups. The droplets (nominal n = 1.33) have a mean DoLP of 0.938, while the 5 and 10 μm borosilicate microspheres (nominal n = 1.47) have mean DoLP values of 0.849 and 0.843, respectively. The small difference between the two borosilicate size groups indicates a comparatively weak size dependence within this specific 5–10 μm comparison. The sand grains (nominal n = 1.50) and soda-lime glass microspheres (nominal n = 1.52) exhibit lower mean DoLP values of 0.797 and 0.763, respectively. Taken together, the group means decrease systematically as the nominal refractive index increases across the investigated particle groups. Because particle material, morphology, and size are not fully controlled independently across all groups, this result is interpreted as an empirical refractive-index-dependent trend rather than evidence that DoLP is determined by refractive index alone.

To quantify the group-level trend shown in Figure 6f, the five mean DoLP values were regressed against the corresponding nominal refractive indices using a first-order least-squares model. The descriptive fit is:

$$\mathrm{D}\mathrm{o}\mathrm{L}\mathrm{P}=-0.852\mathrm{n}+2.080,{\mathrm{R}}^2=0.926$$

(13)

where n is the nominal real refractive index assigned to each particle group. Over the limited range n=1.33–1.52, the fit captures an approximately linear negative association between group-mean DoLP and nominal refractive index. Because the five group means represent only four distinct refractive-index levels—the 5 and 10 μm borosilicate groups share the same nominal refractive index—and because DoLP can also depend on particle size, shape, orientation, absorption, and optical collection geometry, Equation (13) should be regarded as a descriptive fit to the present dataset rather than as a universal DoLP-to-refractive-index calibration. These additional factors are examined further in the T-matrix analysis and discussion below.

5. T-Matrix Simulation and Discussion

To examine the physical plausibility of the experimentally observed DoLP dependence, we used the T-matrix method to calculate polarization-dependent particle scattering [37]. The method expands the incident and scattered electromagnetic fields in vector spherical wave functions and relates their expansion coefficients through a transition matrix. Although the T-matrix framework is applicable to rotationally symmetric non-spherical scatterers, spherical particles are used here to isolate the influence of particle size and refractive index in a simplified model.

The incident and scattered fields are expressed as:

$${\mathit{E}}^{\mathit{i}\mathit{n}\mathit{c}}\left(\mathit{R}\right)={\displaystyle \sum _{n=1}^{n_{max}}}{\displaystyle \sum _{m=-n}^n}\left[a_{mn}Rg{\mathit{M}}_{\mathit{m}\mathit{n}}\left(kR\right)+b_{mn}Rg{\mathit{N}}_{\mathit{m}\mathit{n}}\left(kR\right)\right]$$

(14)

$${\mathit{E}}^{\mathit{s}\mathit{c}\mathit{a}}\left(\mathit{R}\right)={\displaystyle \sum _{n=1}^{n_{max}}}{\displaystyle \sum _{m=-n}^n}\left[p_{mn}{\mathit{M}}_{\mathit{m}\mathit{n}}\left(kR\right)+q_{mn}{\mathit{N}}_{\mathit{m}\mathit{n}}\left(kR\right)\right],\left|R\right|>r_0$$

(15)

Here, r₀ is the radius of a sphere enclosing the particle. The coefficients p_mn and q_mn of the scattered field are linearly related to the incident-field coefficients a_mn and b_mn through the T-matrix.

$$p_{mn}={\displaystyle \sum _{n^{\text{'}}=1}^{n_{max}}}{\displaystyle \sum _{m^{\text{'}}=-n^{\text{'}}}^{n^{\text{'}}}}\left[{\mathit{T}}_{\mathit{m}\mathit{n}{\mathit{m}}^{\mathit{\text{'}}}{\mathit{n}}^{\mathit{\text{'}}}}^{11}{a}_{m^{\text{'}}{n}^{\text{'}}}+{\mathit{T}}_{\mathit{m}\mathit{n}{\mathit{m}}^{\mathit{\text{'}}}{\mathit{n}}^{\mathit{\text{'}}}}^{12}{b}_{m^{\text{'}}{n}^{\text{'}}}\right]$$

(16)

$$q_{mn}={\displaystyle \sum _{n^{\text{'}}=1}^{n_{max}}}{\displaystyle \sum _{m^{\text{'}}=-n^{\text{'}}}^{n^{\text{'}}}}\left[{\mathit{T}}_{\mathit{m}\mathit{n}{\mathit{m}}^{\mathit{\text{'}}}{\mathit{n}}^{\mathit{\text{'}}}}^{21}{a}_{m^{\text{'}}{n}^{\text{'}}}+{\mathit{T}}_{\mathit{m}\mathit{n}{\mathit{m}}^{\mathit{\text{'}}}{\mathit{n}}^{\mathit{\text{'}}}}^{22}{b}_{m^{\text{'}}{n}^{\text{'}}}\right]$$

(17)

Equations (16) and (17) can be written compactly as Equation (18):

$$\left[\begin{array}{c}p\\ q\end{array}\right]=\mathit{T}\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{cc}{\mathit{T}}^{11}& {\mathit{T}}^{12}\\ {\mathit{T}}^{21}& {\mathit{T}}^{22}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]$$

(18)

The T-matrix is then written as:

$$T=-\mathit{R}\mathit{g}\mathit{Q}{\left[\mathit{Q}\right]}^{-1}$$

(19)

Here, Q and RgQ are 2 × 2 block matrices constructed using the corresponding regular and outgoing vector spherical wave functions.

The polarization transformation of the scattered field can be represented by a 4 × 4 Mueller scattering matrix. If S denotes the incident Stokes vector and S’ the scattered Stokes vector, their relationship is:

$${\mathit{S}}^{\mathit{\text{'}}}=\mathit{M}\mathit{S}$$

(20)

For a rotationally symmetric particle under the symmetry assumptions used here, the scattering matrix can be written in the following form:

$$\mathit{M}\left(\mathit{\theta }\right)=\left[\begin{array}{cccc}{\mathit{F}}_{11}& {\mathit{F}}_{12}& 0& 0\\ {\mathit{F}}_{12}& {\mathit{F}}_{22}& 0& 0\\ 0& 0& {\mathit{F}}_{33}& {\mathit{F}}_{34}\\ 0& 0& {-\mathit{F}}_{34}& {\mathit{F}}_{44}\end{array}\right]$$

(21)

The PRDH experiment primarily detects the transmitted reference field together with low-angle forward-scattered light. The T-matrix calculation was therefore used to examine the forward-scattering polarization response. The simulations used spherical, non-absorbing particles with a wavelength of 0.50 μm, refractive-index real parts of 1.33, 1.44, 1.50, and 1.52, and zero imaginary part. Particle-size distributions were represented using the Hansen-Travis model with equivalent-sphere radii of 1, 1.5, 2, 2.5, and 3 μm, an effective variance of 0.10, and seven Gaussian integration points. Because the simulation wavelength and refractive-index grid are not identical to every experimental condition, the comparison is used to examine qualitative trends rather than direct quantitative agreement.

Figure 7a shows the simulated DoLP as a function of scattering angle for particles of different sizes at a fixed refractive index. In the forward-scattering region, the curves for 2-6 μm particles remain relatively close, indicating a weaker size dependence than the refractive-index dependence observed in Figure 7b. At a fixed particle size, increasing the refractive index from 1.33 to 1.52 produces a systematic decrease in DoLP over the low-angle forward-scattering region. This trend is qualitatively consistent with the experimental group-level behavior and supports the physical plausibility of refractive-index-sensitive DoLP measurements. The simulation does not model the complete transfer function of the experimental system, including the exact angular weighting of the telecentric collection optics and the holographic interference process, and is therefore not used as a quantitative calibration of the measured DoLP.

The observed refractive-index sensitivity is consistent with previous Stokes and multi-angle polarization-scattering studies in which refractive index and composition influence the measured polarization response [11,12]. The distinctive feature of the present sensing approach is that this polarimetric information is associated with the same individual particles for which three-dimensional position and morphology are obtained by holographic reconstruction. Several limitations should nevertheless be stated. First, only four distinct nominal refractive-index levels are represented, and the particle groups are not fully matched in material, size, and morphology; consequently, Equation (13) is a descriptive relationship for the present dataset rather than a universal calibration curve. Second, the current transmission/forward-scattering geometry is best suited to transparent or weakly absorbing particles. For opaque or strongly absorbing particles, attenuation of the transmitted field can reduce hologram contrast and increase uncertainty in Stokes retrieval. Third, a quantitative refractive-index inversion would require a calibrated model that includes particle size, shape, absorption, and the angular response of the collection optics. These issues define the scope of the present study and motivate future validation using size-matched particles with a denser refractive-index series and alternative collection geometries.

6. Conclusions

This study presents a polarization-resolved digital holographic sensing approach for particle-resolved multiparameter characterization in three-dimensional particle fields. By combining numerical holographic refocusing with division-of-focal-plane polarimetry, the system associates nine scalar morphological descriptors (x, y, and z coordinates; equivalent diameter; major and minor axes; perimeter; area; and shape factor) with four linear-polarization descriptors (S₀, S₁, S₂, and DoLP) for each resolved particle. Measurements of three standard glass-microsphere groups yielded an average absolute relative deviation of approximately 5.5% between group-mean measured and nominal diameters. Across the investigated particle groups, the group-mean DoLP decreased with increasing nominal refractive index, while the 5 and 10 μm borosilicate groups showed only a small difference in mean DoLP. A descriptive regression of the five group means gave an approximately linear negative trend over n = 1.33-1.52, and T-matrix calculations showed a qualitatively consistent refractive-index dependence in forward scattering. These results support PRDH as an optical sensing framework that co-registers particle morphology and polarization information. The present DoLP-refractive-index relationship should be interpreted as an empirical trend under the tested conditions, not as a universal refractive-index calibration, and further validation with independently controlled particle properties is required for quantitative inversion.