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This work quantifies the polarization persistence and memory of circularly polarized light in forward-scattering and isotropic (Rayleigh regime) environments; and for the first time, details the evolution of both circularly and linearly polarized states through scattering environments. Circularly polarized light persists through a larger number of scattering events longer than linearly polarized light for all forward-scattering environments; but not for scattering in the Rayleigh regime. Circular polarization’s increased persistence occurs for both forward and backscattered light. The simulated environments model polystyrene microspheres in water with particle diameters of 0.1 μm, 2.0 μm, and 3.0 μm. The evolution of the polarization states as they scatter throughout the various environments are illustrated on the Poincaré sphere after one, two, and ten scattering events.
© 2015 Optical Society of America
1. Introduction
In this work, the evolution of circularly and linearly polarized light as it scatters throughout a variety of scattering environments is fully investigated. In particular, we show circularly polarized light exhibits superior persistence for forward-scattering particle environments. Circularly polarized light’s increased persistence compared to linearly polarized light, often called polarization memory, is of importance for many sensing techniques in scattering environments but until now this scattering evolution has not been detailed. This work presents simulation results showing the evolution of polarized light in scattering environments of forward and isotropically (Rayleigh regime) scattering particles for both forward and backscattered photons.
The use of polarized light in scattering environments, specifically the difference between circularly and linearly polarized light’s interactions in scattering environments, is of interest. Previous research has shown benefits for the use of both linear and circular polarization for sensing in specific scattering environments, often showing increased performance benefits for circularly polarized techniques [1–6 ]. Previous work has focused on isolated visible wavelengths and underwater scattering environments. We recently published unique simulation results showing superior persistence for circular polarization versus linear polarization in scattering environments of fog and dust over broad wavelength ranges at infrared wavelengths [7–9 ]. Sensing in scattering environments utilizing polarization relies on the polarization persistence, or polarization memory, of the light. The mechanism of circular polarization’s increased persistence has been theoretically hypothesized; it is proposed that circular polarization depolarizes due to the randomization of the photon’s direction and the randomization of the helicity [10, 11]. Xu and Alfano find the benefits for circular polarization are greatest for large particles with refractive indices close to the air environment (refractive index ∼ 1) and for small, high-index contrast particles (refractive indices between 1.5 and 2) [11]. Xu and Alfano’s work, as well as Bicout et al.’s preceding work, focuses on the analytical uncoiling length and circular depolarization length as their metrics [12]. These works do not present how polarized light evolves in scattering environments toward this increased persistence for circular polarization.
A number of groups have investigated the temporal response of polarized light pulses in scattering environments. Ishimaru et al. performed simulations for polarized pulses of light incident on a slab of latex scattering particles in water for an angularly limited exit beam at a wavelength of 0.53 μm [13]. The solution’s size distribution was nearly monodisperse with a mean particle diameter of 2.019 μm. Their results show circular polarized light has a larger Degree of Polarization (DoP) over time for photons exiting the slab into a narrow forward half angle of 3 degrees. As in this work, Ishimaru et al. show that the DoP for circularly polarized pulses decreases more gradually than it does for linearly polarized pulses for the forward direction. Ishimaru’s work shows circular polarized pulses maintain their DoP longer in time than linearly polarized pulses transmitted through the slab but offers no insight in how or when the polarized states are modified throughout the environment. Kim and Moscoso simulated the temporal variations of backscattered flux for incident circularly polarized pulses [14]. They investigated scattering environments of latex spheres with monodisperse particle distributions of 0.076 μm, 0.189 μm, and 0.303 μm with an illumination wavelength of 633 nm. For the smallest particle size, the dominant backscattered flux is from the opposite handedness circular polarization. For the larger two particle sizes, the initial backscattered state is the opposite handedness but almost immediately the backscattered flux is dominated by the initial incident handedness. Finally, Cai et al. show temporal results for simulations of particle diameters of 0.1 μm, 0.213 μm, 0.855 μm and 8 μm and experiments with particle diameters of 0.213 μm and 8 μm, all at an illuminating wavelength of 610 nm [15]. They generally conclude that circularly polarized pulsed light, of the same handedness dominates backscattered light when the scattering particles are larger than the incident wavelength. When the scattering particles are smaller than the wavelength (Rayleigh regime) the opposite is shown to be the case. These works are all limited to the temporal variation of backscattered light.
Finally, researchers are investigating depolarization and enpolarization, from various scattering processes including scattering from rough surfaces, disordered media, speckle patterns, inhomogeneities, and other complex media [16–22 ].
To date, no research has examined the evolution of incident polarized light, as a function of scattering event for both forward and backscattered photons. In this work we present simulation results for scattering environments of polystyrene microspheres in water. Specifically, we look at monodisperse particle distributions with particle diameters of 0.1 μm, 2.0 μm, and 3.0 μm at an incident wavelength of 543.5 nm. These parameters correspond to isotropic scattering (0.1 μm Rayleigh regime) as well as forward-scattering particles (2.0 μm and 3.0 μm). The parameters are representative of radiation and advection fog at infrared wavelengths. This work was initiated with our recent conference paper which was limited to a single forward-scattering size parameter [23]. Monte Carlo simulations, presented here, for these scattering environments illustrate the evolution of circularly and linearly polarized incident light using the Poincaré sphere after successive scattering events throughout the scattering environment. Circular polarization persists through a larger number of scattering events for both forward and backscattered photons for all of the large particle scattering environments. These results quantify circular polarization’s smooth and slow degradation as a function of scattering event, compared to linear polarization’s abrupt degradation through a scattering environment.
This article is organized as follows: Section 2 covers the background of polarization, polarization-tracking Monte Carlo simulations, and the representation of polarization on the Poincaré sphere, Section 3 presents the Monte Carlo simulation results for linearly polarized light scattering in forward and isotropic scattering environments, Section 4 presents Monte Carlo simulation results for circularly polarized light in the isotropic and forward-scattering environments, comparing these results to those presented for linearly polarized light, and Section 5 concludes, showing circular polarization persists superiorly for forward and backscattered photons in all of the forward-scattering environments.
2. Background
The fraction of measured light that is purely polarized is the Degree of Polarization (DoP). The DoP is defined using the Stokes parameters (S 0, S 1, S 2, S 3) [24],
The DoP ranges from 0 for completely unpolarized light to 1 for purely polarized light. A Stokes vector with a DoP between 0 and 1 is partially polarized. The Poincaré sphere can represent any possible polarization state or Stokes vector [24, 25]. For example, vertical linear polarization is indicated on the Poincaré sphere in Fig. 1 with an orange sphere.
Fig. 1 Poincaré Sphere representation of vertical linearly polarized incident light. (Orange sphere represents location.)
The goal of this work is to investigate the evolution of circularly and linearly polarized light’s polarization state in scattering environments. To this end, we use a polarization-tracking Monte Carlo program for all our simulation results [26]. Both circularly and linearly polarized photons (one million of each polarization state for each simulation) are propagated perpendicular to the face of a slab of scattering media. Rigorous Mie scattering theory is utilized for each simulated scattering event. Scattering particles are modeled as homogenous refractive index (1.597 for polystyrene) spheres. For all the individual scattering events, the polarization and angular scattering properties of a single scattering event are calculated for all incident polarization states and angles [27]. Two parameters of the scattering particles are used within the calculations, the size parameter and the relative refractive index. The relative refractive index is the ratio of the particle’s refractive index to the refractive index of the external medium. The size parameter of the scattering particle is a ratio of the particle’s size and index, to the incident wavelength. In Equation 2, x is the size parameter, a is the radius of the scattering particle, n is the real refractive index of the external medium (not the particle medium), and λ 0 is the vacuum wavelength of the light.
The particle diameters 0.1 μm, 2.0 μm, and 3.0 μm and the incident wavelength of 543.5 nm used in these simulations correspond to unitless size parameters 0.77, 15.4, and 23.1. Results for a particle size of 1.0 μm, corresponding to a size parameter of 7.7, can be found in our recent conference paper [23]. Scattering environments with size parameters in these ranges encompass a wide range of natural environments, such as radiation and advection (marine) fog. These particle sizes and the resulting scattering environments cover both isotropically (Rayleigh regime) and forward-scattering situations.The Monte Carlo simulation tracks the location and polarization state of each photon. Tracking is performed for the location of each photon before and after each scattering event as well as for the polarization state of each photon before and after each scattering event. An initial Stokes vector sets the photon’s initial polarization state. The polarization state of each photon is modified and its Stokes vector is updated after each scattering event. It is important to note, the Stokes parameters of individual photons in the Monte Carlo simulation are single instantaneous “independent light streams” and are thus purely polarized (DoP = 1) [28, 29]. The ensemble of the individual photons Stokes parameters, called the cumulative Stokes state throughout this paper, represents the measurable polarization state and can be purely polarized, partially polarized, or completely unpolarized (0 ≤ DoP ≤ 1).
Forward-scattered photons are those that are scattered to a location further into the scattering slab than the scattering event location. Conversely, backscattered photons are those that are scattered to locations closer to the input face of the slab. Each scattering event is considered independently of previous scattering events. Regardless of previous scattering directions, after the specified scattering event, if the photon is scattering further into the slab it is considered forward scattered; otherwise, if it scatters back toward the input face it is considered backscattered. The scattering direction after the specific scattering event is the only direction used. The density of particles for each simulation was such that a sufficient number of scattering events would occur for each incident photon, resulting in an optical thickness of 10 [7, 27].
3. Linearly polarized initial illumination
The vertical linearly polarized state of the incident photons is plotted on the Poincaré sphere in Fig. 1. The initial Stokes state of the incident photons is located at the center of the large orange sphere. All one million photons for this polarization state started in this position on the Poincar sphere.
In order to reduce the clutter of the plots in the following figures, we choose the first one hundred thousand photons from the incident one million for each plot. The Stokes parameters for each of the one hundred thousand photons are plotted on the Poincaré sphere after a specified number of scattering events. Forward-scattered photons are colored red and backscattered photons are colored blue in the following figures. All the photons’ Stokes parameters are ultimately transformed into the global reference frame set by the initial slab geometry and the initial polarization states. The following figures illustrate the resulting scattered Stokes parameters after 1, 2 and 10 scattering events for each environment. Each individual photon’s Stokes polarization state is plotted on the surface of the Poincaré sphere since they are purely polarized. The cumulative Stokes state, for all the forward or backscattered photons of the incident one million photons, is shown as a large sphere in either orange or purple, respectively. The cumulative Stokes state is located inside or very close to the surface of the Poincaré sphere since it is generally partially polarized.
The following results are split into two sets: 1) for the forward-scattering particle environments, and 2) for the isotropic (Rayleigh regime) scattering particle environment.
3.1. Forward-scattering environments: linear polarization
The normalized angular scattering for each particle is shown on the polar plots in Fig. 2. These plots illustrate the differences between the angular scattering properties of each particle size. The particle is placed at the center of the plot and the plots show how likely it is, for incident radiation from the 180 degree location, to scatter at each angle. The scattering from radiation perpendicularly polarized to the page is plotted in solid black; a dashed blue curve corresponds to parallel oriented incident radiation. For these forward-scattering particles, the blue and black curves are nearly indistinguishable and the blue curves are not visible on the plots. As is evident in the plots, the probability to scatter backwards is very small for these environments. Less than 1 percent of all the incident photons are backscattered after any single scattering event for the forward-scattering particle environments.
Fig. 2 Scattering profiles for particle sizes of (a) 2.0 μm and (b) 3.0 μm. Perpendicular and parallel incident polarization states scattering are plotted as black and blue curves. For these forward-scattering particles, the blue and black curves are indistinguishable and the blue curves are not visible on the plots.
Figures 3 and 4 show the results for incident linearly polarized scattered photons after one (Figs. 3(a) and 4(a)), two (Figs. 3(b) and 4(b)), and ten (Figs. 3(c) and 4(c)) scattering events for the forward-scattering particle sizes 2.0 μm (Fig. 3) and 3.0 μm (Fig. 4). After one scattering event (Figs. 3(a) and 4(a)), forward-scattered photons (red) for the particles remain close to their initial location on the Poincaré sphere. The forward-scattered Stokes parameters initially spread equally along the equator and in the direction of the poles for both particle sizes. The cumulative Stokes state for the forward-scattered photons (orange) is nearly purely polarized for the each of the particle sizes. The backscattered photons Stokes parameters (blue) are spread around the Poincaré sphere. As the particles size increases more of the backscattered photons gain ellipticity. This is especially evident when comparing the results presented here and those in our previous conference paper [23]. Although there are more elliptical states for increasing sizes, the cumulative backscattered photons’ Stokes state (purple) is highly depolarized and has little to no ellipticity. The resulting cumulative backscattered depolarized state is close to the origin of the Poincaré sphere it still has a small preference for vertical linear polarization. Vertical linearly polarized light tends to remain nearly purely polarized if forward scattered after one scattering event. After a single scattering event, backscattered photons are highly depolarized. Both the forward and backscattered photons’ Stokes parameters are spread around the Poincaré sphere more after two scattering events than after one scattering event. After two scattering events (Figs. 3(b) and 4(b)), the cumulative forward-scattered Stokes state is still highly polarized. This state remains close to the initial polarization’s location on the Poincaré sphere. It is hard to see in the figures but the cumulative backscattered Stokes state is slightly more depolarized than after one scattering event. Forward-scattered photons’ Stokes parameters for both particle sizes spread more around the equator than toward the poles. After ten scattering events (Figs. 3(c) and 4(c)) the individual photons’ polarization states are highly spread around the Poincaré sphere. Backscattered Stokes parameters increasingly spread around the Poincaré sphere as the particle size grows. The cumulative backscattered state is located near the origin, DoP ∼ 0, nearly completely depolarized for both particle sizes. Although it is not visible in the figures, the cumulative forward-scattered state is inside the surface of the Poincaré sphere and has depolarized. The depolarization of the vertical linearly polarized forward-scattered photons tends to spread along the equator more than toward the poles. This preference to spread along the equator is present for all the forward-scattering particle sizes. For the forward-scattering environments, linear polarization depolarizes into other linear polarization states faster and more readily than into elliptical polarization states.
Fig. 3 Scattered Stokes parameter values for incident linearly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 2.0 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
Fig. 4 Scattered Stokes parameter values for incident linearly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 3.0 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
3.2. Isotropically scattering environment (Rayleigh regime): linear polarization
The normalized angular scattering for the 0.1 μm particle is shown on the polar plots in Fig. 5. This plot is similar in design to the plots in Fig. 2. However, for the isotropic scattering particle, the parallel and perpendicular polarizations scattering are distinct. The isotropic scattering particle has a much larger amount of backscattering compared to the forward-scattered particles. After any single scattering event, roughly 42 percent of the incident photons will be backscattered for the isotropically scattering environment.
Fig. 5 Scattering profile for a particle size of 0.1 μm. Perpendicular and parallel incident polarization states scattering are plotted as a solid black and dashed blue curves.
Figure 6 shows the results for incident linearly polarized scattered photons after one (Fig. 6(a)), two (Fig. 6(b)), and ten (Fig. 6(c)) scattering events for the isotropic scattering particle size 0.1 μm. The results for the isotropic scattering environment are drastically different than the forward-scattering environments. After the first scattering event (Fig. 6(a)), forward (red) and backscattered (blue) photons remain along the equator of the Poincaré sphere, thus linearly polarized states. The cumulative forward (orange) and backscattered (purple) Stokes states are slightly depolarized for the isotropic scattering environment. Although there appear to be a multitude of states around the equator, the cumulative states show the majority of the scattered photons remain near the initial location on the Poincaré sphere but have moved just inside the surface of the sphere, DoP < 1. After two scattering events (Fig. 6(b)), the photons Stokes parameters remain near the equator but the cumulative forward and backscattered Stokes states are more depolarized and move toward the center of the Poincaré sphere. Both the cumulative forward and backscattered Stokes states depolarize along the S 1 axis. After ten scattering events (Fig. 6(c)), the forward and backscattered states are evenly distributed around the equator of the Poincaré sphere. The cumulative forward and backscattered Stokes state are both at the origin and thus completely depolarized, DoP ≈ 0.
Fig. 6 Scattered Stokes parameter values for incident linearly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 0.1 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
4. Circularly polarized initial illumination
The incident right circularly polarized state for this next set of simulations is located at the positive z axis of the Poincaré sphere. The incident photon’s initial polarization state is plotted on the Poincaré sphere in Fig. 7. The initial Stokes state is located at the center of the large orange sphere. All one million photons for this circular polarization state started in this position on the Poincaré sphere.
Fig. 7 Poincaré sphere representation of right circularly polarized incident light. (Orange sphere represents location.)
As in the previous linear polarization case, we chose the first one hundred thousand photons, from the incident one million, for each of the following plots. The same plotting scheme is utilized: red for individual forward-scattered photons, blue for individual backscattered photons, the orange sphere represents the cumulative Stokes state for forward scattering, and the purple sphere represents the cumulative Stokes state for backscattering.
4.1. Isotropically scattering environment (Rayleigh regime): circular polarization
Figure 8 shows the results for incident circularly polarized scattered photons after one (Fig. 8(a)), two (Fig. 8(b)), and ten (Fig. 8(c)) scattering events for the isotropic scattering particle size 0.1 μm. After the first scattering event (Fig. 8(a)), the forward (red) scattered photons maintain their right handedness but are spread around the entire positive S 3 hemisphere. Conversely, the backscattered (blue) photons reverse handedness and spread around the entire negative S 3 hemisphere. Both forward and backscatter states modify to a plethora of polarization states. Although it is not apparent from the plot, the cumulative forward (orange) and backscattered (purple) Stokes states are only somewhat depolarized. The cumulative states are on the S 3 axis but have moved just below the surface of the Poincaré sphere. The backscattered cumulative state has also flipped handedness. After two scattering events (Fig. 8(b)), the forward and backscattered photons Stokes parameters are intermixed on the entire Poincaré sphere. The forward and backscatter photons are no longer clearly separated by handedness. The cumulative forward and backscattered Stokes states are now highly depolarized. The cumulative forward-scattered Stokes state is still right-handed and the cumulative backscattered Stokes state is still left-handed. After ten scattering events (Fig. 8(c)), both forward and backscattered photons have migrated toward the equator and have lost most of their handedness. The cumulative forward and backscattered Stokes states are completely depolarized and located at the origin, DoP = 0. As the number of scattering events increases, circularly polarized photons evolve into a collection of linearly polarized states, resulting in completely depolarized cumulative forward and backscattered states.
Fig. 8 Scattered Stokes parameter values for incident circularly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 0.1 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
The plots in Fig. 6 and Fig. 8 show linear and circular polarization’s modification due to the isotropically scattering environment after individual scattering events. The evolution of the cumulative forward and backscattered DoP plots, as a function of scattering event, are shown in Fig. 9.
Fig. 9 Cumulative DoP, for forward (x’s) and backscattered (o’s) photons, from circularly (red) and linearly (black) polarized incident polarization states versus number of scattering events. Both linear and circular forward and backscattered photons depolarize rapidly as a function of scattering event. Circularly polarized light is completely depolarized after merely eight scattering events while linear polarization is completely depolarized after fourteen scattering events.
The initial DoP for forward or backscattered photons in this plot is set to 1, purely polarized. Even though linear polarization is superior for this isotropic scattering environment, both linear and circular forward and backscattered photons depolarize rapidly as a function of scattering event. Circularly polarized light is completely depolarized after merely eight scattering events while linear polarization is completely depolarized after fourteen scattering events.
4.2. Forward-scattering environments: circular polarization
Figures 10 and 11 show the results for incident circularly polarized scattered photons after one (Figs. 10(a) and 11(a)), two (Figs. 10(b) and 11(b)), and ten (Figs. 10(c) and 11(c)) scattering events for the forward-scattering particle sizes 2.0 μm (Fig. 10) and 3.0 μm (Fig. 11). The forward-scattering environments for both the linear as well as circular incident polarizations exhibit a very different behavior than the isotropic case. For this incident circular polarization, after the first scattering event (Figs. 10(a) and 11(a)), forward-scattered photons remain close to their initial pole location on the Poincaré sphere. The forward-scattered photons’ cumulative Stokes state is nearly purely polarized, DoP ≈ 1. The backscattered photon’s Stokes parameters are spread out around the Poincaré sphere more than the forward-scattered photons, but the Stokes parameters remain in the same handedness. As the forward-scattering particle size increases the backscattered Stokes parameters fill more of the hemisphere. The resulting cumulative backscattered Stokes state is depolarizing. Circularly polarized light tends to remain nearly purely polarized if forward-scattered but is slightly depolarized if backscattered. Remember that after one scattering event, just over 1 percent of the incident photons are backscattered for each of the forward-scattering particle environments as was illustrated in Fig. 2. After two scattering events (Figs. 10(b) and 11(b)), the photons’ Stokes parameters for both forward and backscattering are spread around the Poincaré sphere slightly more so than after one scattering event. The cumulative forward-scattered Stokes state for all the forward-scattering particle environments remain close to the initial polarization’s location and is still nearly purely polarized. The forward-scattered photons’ Stokes parameters remain in a cap packed near the R pole, spreading down from the pole location. The cumulative backscattered Stokes state is slightly more polarized after two scattering events than after one scattering event. Photons that are backscattered largely maintain right-handed helicity but fill more of the hemisphere than after one scattering event. After ten scattering events (Figs. 10(c) and 11(c)), the forward-scattered photons’ Stokes parameters are spread out on the Poincaré sphere more so than after two scattering events. Even with this increased spreading, the Stokes parameters remain in a cap near the R circular pole and are still highly polarized. The backscattered Stokes parameters are more spread out and nearly fill the entire upper hemisphere. Although it is not visible in the figures, the cumulative forward-scattered Stokes state is located just under the R pole cap and is highly polarized. The cumulative backscattered Stokes state is also highly polarized. Circularly polarized incident light maintains a high DoP, for both forward and backscattered photons, through a large number of scattering events for the forward-scattering environments.
Fig. 10 Scattered Stokes parameter values for incident circularly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 2.0 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
Fig. 11 Scattered Stokes parameter values for incident circularly polarized light after (a) 1, (b) 2, and (c) 10 scattering events for a scattering environment consisting of particles with diameter 3.0 μm and an illuminating wavelength of 543.5 nm. This figure shows the first 100,000 photons’ resulting Stokes parameters after each scattering event; forward-scattered photons are shown in red and backscattered photons are shown in blue. The resulting cumulative Stokes state, for the forward or backscattered photons, is shown as large orange or purple spheres.
The plots in Fig. 3, Fig. 4, Fig. 10, and Fig. 11 show linear and circular polarization’s modification due to the forward-scattering environments after individual scattering events. The evolution of the cumulative forward and backscattered DoP plots, as a function of scattering event, are shown in Fig. 12.
Fig. 12 Cumulative DoP, for (a) backscattered and (b) forward-scattered photons, from circularly (red) and linearly (black) polarized incident polarization states versus number of scattering events. The two particle sizes are plotted as follows: 2.0 μm is plotted with stars and 3.0 μm is plotted with triangles. Forward and backscattered light from incident circularly polarized light for the forward-scattering environments maintains its DoP and therefore persists through a larger number of scattering events.
Figure 12 clearly shows that circular polarization maintains its DoP better than linear polarization. For the forward-scattering environments, circular polarization persists through increasing scattering events better than linear polarization for either forward or backscattered photons. In Fig. 12, the initial DoP for forward or backscattered photons is set to 1, purely polarized. In Fig. 12(a), after only one scattering event backscattered linear polarized light is highly depolarized. Backscattered circular polarized light is also depolarized after one scattering event, although not to the extent of linear polarization. As the number of scattering events increases, linearly polarized light’s DoP decreases to a completely unpolarized state. Conversely, backscattered circular polarized light’s DoP increases as the number of scattering events increases. After about ten scattering events, circular polarized backscattered light peaks to its largest DoP. After this, circular backscattered photon’s DoP begins to decrease but remains highly polarized even after 30 scattering events. It is difficult to understand the significance of this initial depolarization and then increase in DoP for backscattered circular polarized photons from the forward-scattering environments since the relative number of backscattered photons is so small. After one scattering event just over 1 percent of the light is backscattered for these forward-scattering environments. After successive scattering events the number of backscattered photons increases and there are a larger number of photons for the ensemble cumulative Stokes state. In Fig. 12(b), circularly incident forward-scattered photons remain highly polarized through a larger number of scattering events. The linearly polarized incident photons remain highly polarized for small numbers of scattering events but depolarize more quickly than circularly polarized photons. Circular polarization remains largely polarized as the number of scattering events increases while linear polarization depolarizes quickly by evolving into a plurality of linearly polarized states. Overall, forward and backscattered light from incident circularly polarized light for the forward-scattering environments maintains its DoP and therefore persists through a larger number of scattering events.
5. Conclusions
This work quantitatively and qualitatively presents the evolution of linear and circularly polarized light as it scatters throughout both isotropic (Rayleigh regime) and forward-scattering environments. Circularly polarized light persists through a larger number of scattering events longer than linearly polarized light for all forward-scattering environments. In this forward-scattering environment, circular polarization’s increased persistence occurs for both forward and backscattered light. The simulated forward-scattering environments modeled polystyrene microspheres in water with particle diameters of 2.0 μm and 3.0 μm. The scattering profiles of these environments are consistent with advection (marine) and radiation fog at infrared wavelengths. The evolution of the polarization states as they scatter throughout the various environments are illustrated on the Poincaré sphere after one, two, and ten scattering events.
We also model a more isotropically scattering environment with a 0.1 μm particle diameter. In this isotropic scattering regime circularly and linearly polarized light depolarize rapidly. Linear polarization persists better for this isotropic environment. Incident linearly polarized light depolarizes into various other linearly polarized states while incident circularly polarized light depolarizes into a multitude of elliptical states initially and then evolves into more and more linear polarized states.
For all of the forward-scattering environments, circular polarization maintains a high degree of polarization, and remains in a range of states near the incident polarization state, throughout a large number of scattering events. Linear polarization depolarizes more rapidly into other linearly polarized states than into elliptical states, leading to a more highly depolarized cumulative state compared to that of circular polarization. This work shows clearly that circular polarization is superior to linear polarization in maintaining its DoP as a function of scattering event, and persisting through the larger forward-scattering particle environments. Circularly polarized light slowly, and smoothly degrades from its initial state, maintaining the same handedness, while linearly polarized light abruptly depolarizes into a plethora of other linear polarization states. This work quantifies the polarization persistence and memory of circularly polarized light in forward-scattering environments; and for the first time, details the evolution of both circularly and linearly polarized states through scattering environments.
Acknowledgments
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy (DOE) National Nuclear Security Administration under contract DE-AC04-94AL85000.
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