1. Introduction

The optical scattering effect of the medium has negative influence on the optical detection, identification and imaging of targets [1–3]. Turbid water, smoke and dense fog are usually considered as strong scattering media because containing abundant randomly-distributed scattering particles [4]. The scattering of light through these media is generally divided into single scattering and multiple scattering. The single scattering light still carries partial information of the initial incident light and makes dominant contribution to the forward scattering while the multiple scattering light losses most of the carried information and increases the background noise greatly. For sparse medium, the single scattering theory is more available. For dense medium, while multiple scattering occurs often and the diffusion model can be used to describe this process [5]. Both of the single and the multiple scattering have great influence on the measurement of light transmission. Fog is the most common scattering medium affecting outdoor imaging and optical communications. Scattering is more important than absorption for fog in the waveband ranging from visible to near infrared. So, it is of great significance to build scattering models of light scattering through fog in broad band spectrum [6,7].

Because the scattering processes are random and complex, it holds great challenge to investigate the scattering properties quantitatively, let alone to derive a brief formulation to describe the internal physical properties in a medium theoretically. In previous works, researchers focused on the influence of particle species, morphology and the variation of wavelength on the single and multiple scattering characteristics based on some qualitative theoretical models and empirical models [8–13]. However, theoretical derivation could not alone prove the reliability of the models, so a series of Monte Carlo simulation methods were utilized to prove the proposed theoretical models [14–19]. The fact that Monte Carlo simulations only deal with case-by-case situations limits the application range of the models. Hence, some actual experimental measurements were very required to verify the applicability of the models. Some works had analyzed the off-axis scattered radiation experimentally as well as theoretically. Further, the radiative transfer equations were approximately derived based on the small-angle approximation and a diffusion theory. But those equations containing some parameters need to be measured or simulated were too complicated to clearly exhibit the natural properties of the medium [20]. Overall, more experiments about the scattering characteristics have not been conducted combining with theories and simulations up to now. But this combination is very desired for a deep understanding about the optical properties of scattering medium.

In this work, we developed two quantitative theoretical models to describe the single scattering and the multiple scattering of narrow beams through fog based on the different polarization properties of these two scattering processes on the condition of paraxial approximation. According to a forward scattering theory that the intensity of single scattering is much stronger than that of multiple scattering within a small scattering angle [14], the single scattering light can be approximately replaced by the total scattering light. Moreover, nearly all the scattering light with a polarization direction perpendicular to the that of incident light comes from the multiple scattering because the forward single scattering light still keeps the polarization state of incident light. Hence, the multiple scattering light can be effectively discerned from the total scattering light. In addition, the light scattering processes were simulated with a Monte Carlo algorithm as previous works [21,22]. Meanwhile, the intensities of different scattering light and the variation of the depolarization ratio were also measured over a broad-band spectrum. The results showed both of the experimental measurements and simulation results were in great consistent with the quantitative theoretical models. So the established scattering models were demonstrated to be accurate and universally effective.

2. Theoretical model

2.1 Characterization of optical depth

Here, a parameter of optical depth was used to describe the concentration of fog which was produced by an ultrasonic water mist generator. The intensity of the incident light I₀ is invariable and the intensity of the transmission light I can be measured in real time by a power meter. So, the optical depth τ can be expressed by Beer’s law, which is

The distance is invariable in the experimental system, so the optical depth, τ, is a function of particle concentration ρ. There is a linear relationship between the optical depth and the particle concentration. According to Eq. (1), the transmission light is:

2.2 Single scattering

When resolving the intensity of the single scattering, a mathematical integration approach was proposed under the condition of the paraxial approximation. For an off-axis detector located several beam radii from the optical axis, the optical path length of single scattering light reaching here was approximately equal to the point at axis. Considering a volume differential with thickness of dx as Fig. 1 shows, the scattered light intensity dI(x) (ignoring absorption of fog) can be expressed as:

 

Fig. 1. Integral solution for single scattering.

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The scattering light intensity dI(x) consists of two components: forward scattering light and backward scattering light. Because the scattering phase function is only determined by a particular scattering particle. The intensity of the light scattered by the volume differential and finally reaching the detector, dI_F(x), is proportional to the total scattering light intensity as:

According to Beer’s law, the intensity dI_S of the scattering light collected by the detector at x = L can be expressed as:

The total light intensity I_S collected by the detector can be expressed as the integral over the whole length of the fog:

In Eq. (8), the intensity of scattering light is expressed as a function of optical depth. Therefore, it is easy to derive the maximum intensity at the inflexion point τ = 1.

Within a small scattering angle, the intensity of forward single scattering light is approximately equal to the intensity of total scattering light I_S at the optical axis, so Eq. (8) can be rewritten as:

However, the above approximation is no longer reasonable when the polarization state of scattered light is considered.

2.3 Multiple scattering

In our experiments, the fog was restricted in a box. When the light beam passes through a fog box, the lost energy must be equal to the light energy irradiated from the fog to the clean air (neglect the absorption of the fog) as Fig. 2 shows, which situation is very similar to the thermal diffusion process with a heat source. In addition, almost of the light irradiated into the air experience a multiple scattering process. Hence, the multiple scattering model can be approximately analogized to the diffusion model [14].

 

Fig. 2. Schematic diagram of the light diffusion.

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As Fig. 2 shows, the light intensity of an area element, ds, is I (x, y, z), the gradient of the light intensity at the surface is ∇I (x, y, z), then the light irradiated from this area element can be expressed as C·I·∇I·ds, where C is a diffusion coefficient [14]. The total intensity of multiple scattering light, I_MS, is obtained through an integral over the whole surfaces of the fog box and expressed as:

So, the intensity of the area element is proportional to the total intensity of the multiple scattering light: I (x, y, z) ∝ I_MS.

If the incident light has a horizontal polarization state, then the vertical component of the multiple scattering light I⊥ MS can be expressed as:

For the detected light at the optical axis:

Although we claimed the incident light beams have a horizontal polarization state, it is impossible to generate an absolute linear-polarization light actually. Hence, the incident linear-polarization light contains a very strong horizontal component ${I_{\left\|{}\right.} ^0}$ (${I_{\left\|{}\right.} ^0}$ ≈ I₀) and an extremely weak vertical component I$0 \bot$. The forward scattering theory confirms that the single scattering light still keeps the initial polarization state at the optical axis, so it is inferred theoretically that the light with vertical polarization state on the optical axis only comes from the multiple scattering. But in fact, the measured vertical component at the optical axis consists of the multiple scattering light with vertical polarization state and the transmitted part of the initial vertical component, I$0 \bot$· e^-τ. So, the value of the former is equal to the difference between the totally detected light intensity of the vertical component and the latter:

For the detected light in the off-axis regions:

Although the polarization state of the off-axis single scattering light just varies slightly, the light with vertical polarization state in the off-axis region within a small scattering angle still contains two parts: the vertical components of the multiple scattering light and the single scattering light with a slightly changed polarization state.

So, the sum of the light intensity of the vertical component in the off-axis region is given by:

Because the vertical component intensity of the single scattering light is proportional to the total light intensity of the single scattering: I⊥ SS ∝ I SS, the vertical component I⊥ SS can be expressed as:

Then the total vertical component of the scattering light in the off-axis region with a small scattering angle can be expressed as:

Because the incident light beams had a horizontal polarization state with a very large degree of linear polarization, the intensity of initial incident light, I₀, is approximately equal to ${I_{\left\|{}\right.} ^0}$. So, it is easy to derive that, the maximum value of I⊥ S appears at the point τ = 1+ B₂ / A₂, that is, the vertical component of the scattering light will have an inflection point in the interval τ >1.

2.4 Depolarization ratio

In this paper, a depolarization ratio(also called depolarization degree) was employed to describe the depolarization phenomenon quantitatively, which is a fundamental physical property defined as: the ratio of the light intensity of the vertical component I_⊥ (perpendicular to the electric field direction of the incident light) to the parallel component I_∥ (parallel to the incident light),

Based on the models established in section 2.2 and section 2.3, the expression of depolarization ratio was derived here. At the optical axis, the scattering angle of the single scattering can be approximated as equal to 0°. Therefore, the single scattering light has no change on the polarization state at the optical axis. The detected light intensity of vertical component consists of two parts, the transmitted part of the initial vertical component, I$0 \bot$· e^-τ, and the vertical component I⊥ MS of the multiple scattering light at the optical axis. Then the total vertical component I_⊥ can be expressed as:

According to Beer's law, the light intensity of horizontal components I_∥ detected by the power meter can be expressed as:

An assumption hinted in this derivation should be kept in mind that the horizontal components of multiple scattering was too weak to be ignored compared with ${I_{\left\|{}\right.} ^0}$· e^-τ. Whereas, the exponentially decayed horizontal component will gradually become comparable with the horizontal component of multiple scattering with increasing optical depth. So, the expression of depolarization ratio should be corrected based on the Eq. (19) when the optical depth is too large.

3. Experimental methods and simulations

3.1 Experimental system

The whole measuring system is shown in Fig. 3. Five solid-state lasers with wavelengths of 532 nm, 671 nm, 721 nm, 914 nm and 1645 nm were utilized as the light source (S) respectively. All the diameters of these five light beams were about 2 mm. A variable attenuator (A) controlled the illumination power. The incident polarization states of the collimated light beams were controlled by the polarization state generator (PSG), which was composed of a quarter-wave plate (P) and a rotatable linear polarizer (L1). The fog generation system (FGS) contained two parts: an ultrasonic fog generator and a fog box (0.5 × 0.5 × 1.5 m³). Controlling the spray speed of the fog generator could give rise to different concentrations of fog. A convex lens (CL) was used to collect as much scattering light as possible into the power meter in an off-axis region, which center was 30 mm from the optical axis. Another rotatable linear polarizer (L2) was used to measure the light intensity in different polarization directions. A personal computer (PC) was connected to two light power meters (LPM1, and LPM2) to record optical power.

 

Fig. 3. Schematic of the measuring system for the scattering light.

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The artificial fog environment was generated by the ultrasonic fog generator using pure water. The light power of transmitted light was measured by LPM2 in real time to obtain the values of various optical depth using Eq. (1), while LPM1 measured the scattering light power simultaneously. For the calculation of depolarization ratios, the L2 was moved to the front of the LPM2. Then, the vertical and horizontal components at the optical axis were measured through rotating L2. Finally, the intensities of scattering light in the visible bands (532 nm, 671 nm, 721 nm) were measured and the depolarization ratios of five narrow light beams covering visible-near infrared band (532 nm, 671 nm, 721 nm, 914 nm, 1645 nm) were calculated.

3.2 Simulations

During the simulations, the influence of the air molecules and the impurities is ignored. So, the propagation of polarized light in the FGS was completely affected by fog particles. A Monte Carlo method was applied to simulate the light scattering from the homogeneous particles and the refractive index of these particles was set as 1.33. The wavelength of the incident light was determined by the working light source. The particle size was set as 5 µm as the real fog particle [23]. During the simulation using a Monte Carlo method, it is naturally workable to distinguish the single scattering photons and the multiple scattering photons because recording the number of collisions between photons and particles is easy to implement algorithmically. As definition, the ballistic photons do not experience collision, while the single scattering-photons experience collision only once. Then, the rest are multiple scattering photons. Consequently, no matter at the optical axis or in the off-axis region, above three kinds of photons can be easily counted algorithmically.

4. Results and analysis

The following sections shows the experimental results, simulation results and theoretically fitting results of the single scattering, multiple scattering and depolarization phenomenon in a broad-band spectrum. An impressive fact that the former two set of results were completely matched with the quantitative models fully demonstrated the effectiveness of the proposed theories.

4.1 Single scattering

All the light powers of above laser beams were set different, so, the experimentally measured light intensities were normalized as the ratio I SS / I₀ for the effective comparison with simulation results and fitting results. Figure 4 presents the normalized intensity of single scattering light with wavelengths of 532 nm, 671 nm and 721 nm in an off-axis region.

 

Fig. 4. Normalized intensities of single scattering light with wavelengths of (a) 532 nm, (b) 671 nm, and (c) 721 nm. (d) The simulated single scattering. The red solid lines mean the fitting curves based on the quantitative model.

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In Figs. 4(a), (b), and (c), the experimental data points could be completely fitted using Eq. (8). Figure 4(d) shows the normalized simulation results of single scattering for three narrow light beams. These results were also completely matched with the quantitative models. In order to increase the signal to noise ratio of simulation, the collection area of scattering light during simulation was larger than the real area. As a consequence, the simulated forward scattering efficiencies were slightly larger than the experimentally measured values for the three light beams as Figs. 4(a)-(d) show. However, both of the experiments and simulations showed that the forward scattering efficiency of light with wavelength of 721 nm falls in between another two wavelengths (A532 1 = 0.018, A671 1 = 0.028, A721 1 = 0.024). In other words, the simulated results were qualitatively consistent with the experimental results. What’s the most attractive for Fig. 4 is that all the fitting curves reached the maximum value at a same optical depth τ = 1, just as we expected in section 2.2. To sum up, the experiments, fittings and simulations were combined to demonstrate the scientificity and accuracy of the single scattering model derived in section 2.2. Furthermore, the assumption that the total scattering light intensity is approximately equal to the single scattering light under a condition of paraxial approximation was proved to be reasonable.

4.2 Multiple scattering

To verify the accuracy of the multiple scattering models established in section 2.3, the vertical components of multiple scattering light at the optical axis and in the off-axis region were experimentally measured, algorithmically simulated and theoretically fitted.

Figure 5(a) shows the calculated results by Eq. (12), which results displayed the vertical component of the multiple scattering light with wavelength of 532 nm at the optical axis. Figure 5(b) shows the normalized simulation results of the single scattering and the multiple scattering (ignore the polarization state) at the optical axis. The simulated single scattering still shows the similar behavior as the model expects. The simulated multiple scattering result in Fig. 5(b) shows the same various tendency with the vertical component of multiple scattering in Fig. 5(a). The tendency of approach to saturation with the increasing optical depth had a clear physical interpretation that when the optical depth was large enough the incident light would be converted into multiple scattering light completely and the intensity of multiple scattering light tended to be a constant. What is more important, both the two results were completely consistent with the multiple scattering model even though the simulation results didn’t consider the polarization. So, the proportion of vertical component to the total multiple scattering is verified to be independent with the optical depth. Consequently, the multiple scattering model was demonstrated to be convincing and the assumption implied in the model was reasonable as well.

 

Fig. 5. (a) Calculated vertical component of the multiple scattering light at the optical axis. (b) Normalized simulation results of single scattering and multiple scattering at the optical axis. The red lines mean the fitted curves based on the quantitative models.

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However, Fig. 5(b) also displays a noticeable phenomenon. When the optical depth is beyond 4, the intensity of the multiple scattering will be larger than that of the single scattering. So, it seemed that the assumption about single scattering model (the single scattering light is approximately equal to total scattering light at the optical axis) was not applicable when the optical depth was over 4. In fact, this problem was cleverly solved in experiments. After a multiple scattering, the output direction of a photon is arbitrary when reaching the window of the fog box. Because the detector was about 10 cm from the fog box, only a little multiple scattering light with very small scattering angle could be collected by the detector. As a result, the proposed single scattering model still worked well even the optical depth was larger than 4 in experiments. Finally, as Fig. 5 shows the experimental results were perfectly fitted in a wide range of optical depth (0∼5.5) using the proposed single scattering model.

In the off-axis region, the vertical components of the total scattering light were focused into the LPM1 by a lens. Since the lens was placed close to the output window of the fog box, the multiple scattering light within a large scattering solid angle were also detected by the LPM1. So, the detected light was the sum of the vertical components of the single scattering light and multiple scattering light. The measured intensities of the light with a vertical polarization state were normalized with the intensities of the initial incident light, as Fig. 6 shows. Next, all the experimental data were completely fitted by Eq. (15). This result further demonstrated that the proposed quantitative models were scientific and effective.

 

Fig. 6. Normalized vertical components of the total scattering light with wavelengths of (a) 532 nm, (b) 671 nm, and (c) 721 nm. The red solid lines mean the fitting curves based on the quantitative model. (d) The statistical values of the inflection point for different light beams.

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Figure 6(d) shows the statistical results of several experiments. The average values of inflection points were labeled in the figure. According to the derivative of Eq. (15), it is easy to derive that, the maximum value of I⊥ S appears at the point τ = 1+ B₂ / A₂. The average values of optical depth at which the intensities reached the local maximums were 1.30, 1.20, 1.16 for the scattering light with wavelengths of 532 nm, 671 nm and 721 nm, respectively. These values mean that the values of B₂ / A₂ are 0.30, 0.20 and 0.16 for the scattering light with wavelengths of 532 nm, 671 nm and 721 nm, respectively. There is an experimental phenomenon that the fitted parameter B₂ reduces with increasing wavelength, while in stark contrast the value of A₂ increases with increasing wavelength. So, the relationship between A₂ and wavelength is totally different with the relationship between A₁ and wavelength. The different meanings of A₁ and A₂ hints that the polarization change of forward single scattering is sensitive to wavelength. We believe a complete interpretation about above results could be found out using the Monte Carlo simulation method containing a Mueller matrix.

4.3 Variation of depolarization ratio

The depolarization ratios of the narrow light beams were calculated via measuring the two orthogonal components of total light at the optical axis. The experiments were conducted over the waveband from 532 nm to 1645 nm as Fig. 7 shows. Before the fitting operations, it is necessary to check the applying condition of Eq. (19) that the exponentially decayed horizontal component of the initial light beam ${I_{\left\|{}\right.} ^0}$· e^-τ is much larger than the horizontal component of multiple scattering light ${I^{\left\|{}\right.}_{MS}}$.

 

Fig. 7. Depolarization ratios of light beams through fog. The solid lines mean the fitting curves based on the quantitative model.

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It is easy to calculate that when the optical depth was 5, the horizontal component of initial incident light was reduced to 0.67% of its initial value. Under this optical depth, the output multiple scattering light preferred to become natural light and the vertical component and horizontal component of multiple scattering were approximately equal. So, the horizontal component of multiple scattering was approximately 0.1% of the initial incident light under an optical depth of 5 according to the result in Fig. 5(b). As a consequence, the using condition of Eq. (19) was also satisfied and all the experimental data were completely fitted by Eq. (19). Figure 7 also shows that the depolarization ratio decreases with the increasing wavelength. An equivalent conclusion was that the light beam with long wavelength performs a better polarization-maintaining property. In addition, the parameter B₁ decreased with increasing wavelength, which tendency was consistent with parameter B₂. This same tendency was reasonable in physics because after a multiple scattering a photon will pass through the fog box randomly at the optical axis or in the off-axis region. Theoretically, an enough large optical depth would lead to a saturation value of 1 for the depolarization ratio because the horizontal component of the multiple scattering must be taken into consideration under this condition. Nonetheless, such large optical depth is rare in practical applications, so the expression about the depolarization ratio is applicable always.

5. Conclusions

In this paper, two quantitative physical models were established to describe the processes of single scattering and multiple scattering for broad-band narrow-beam light through fog. These models greatly simplified the previous expressions about optical scattering and also displayed convincing abilities to interpret the experimental results. All the data from experiments and simulations could be perfectly fitted by the theoretical equations. In addition, the fitting parameters kept self-consistent throughout the whole fitting operations. As a result, the models were demonstrated to be effective and applicable for fog. Moreover, the intrinsic optical properties of scattering media become obvious and easy to understand because of the explicit function expressions used in these models. This work provided a simple but convenient strategy to analyze the propagation of light beams in fog, which holds huge potential in the fields of optical communication and imaging.