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Quasi-rayleigh polarization leap of monodisperse spherical particle as a tool to detect particle radius
Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Quasi-rayleigh polarization leap of monodisperse spherical particle as a tool to detect particle radius Dmitry Petrov∗, Elena Zhuzhulina Crimean Astrophysical Observatory (CrAO RAS), Nauchnyj, Crimea 298409, Russia
a r t i c l e
i n f o
Article history: Received 24 May 2019 Revised 16 December 2019 Accepted 16 December 2019 Available online 17 December 2019 Keywords: Mie theory Rayleigh scattering Polarization Polystyrene beads
a b s t r a c t When the size of scattering particle is increased to be comparable with wavelength, Rayleigh approximation becomes not valid. In polarization scattering pattern this quasi-rayleigh behavior manifests itself as a sharp drop of the degree of linear polarization to the negative values. In this paper, we studied the properties and features of this quasi-rayleigh polarization leap for monodisperse spherical scatterers. The main regularities that determine the wavelength position of the quasi-rayleigh polarization leap are established depending on the phase angle, refractive index and size parameter of the scatterers. A method is suggested for determining the radius of spherical scattering particles of the polystyrene beads in Earth’s atmosphere. A simple interpolation formula is given, which allows, based on observational data, such as phase angle and the wavelength at which the quasi-rayleigh polarization leap is observed, to calculate the particle radius. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Rayleigh lightscattering means the scattering of light by enough small objects. Main properties of such kind of scattering were originally described by Rayleigh [1–5] to explain the color and polarization of the light from the sky. More generally, it describes the scattering of light by particles much smaller than the wavelength of the light, and smaller than the wavelength of the light divided by the real part of the complex refractive index m = n + i·k. As well known, the scattering properties of any object depend on the ratio between the size and wavelength of light. For this reason, the scattering particle is characterized by a size paramer ter X = 2π λ , where λ is the wavelength of the incident light, and r in the case of spherical scatterer is the radius of sphere [6]. For Rayleigh scattering, the following condition should be complied: X|n| <<1. When such a small particle is illuminated by electromagnetic waves, every part of particle experiences the electromagnetic field simultaneously. This field results in the charges separation, so the scattering particle acquires a dipole moment, which changes in consistent with the external electromagnetic field of incident light. The particle then radiates scattered light as a result of the accelerating movement of charges. An oscillating dipole produces an oscillating electromagnetic scattered field, which is proportional to the squared frequency of the oscillation. Since the intensity of light is
∗
Corresponding author. E-mail address: [email protected] (D. Petrov).
https://doi.org/10.1016/j.jqsrt.2019.106806 0022-4073/© 2019 Elsevier Ltd. All rights reserved.
proportional to the square of the amplitude of the electric field, the resulting intensity of scattering depends on wavelength as frequency to the forth power, i.e. I ∼ λ−4 . Moreover, the intensity of scattered light depends on the phase angle α , i.e. on the angle between the direction lightsource - scatterer and scatterer - observer. The phase function of Rayleigh scattering depends on the phase angle as:
I∼
1 + cos2 α
λ4
.
(1)
Even when scattering particle is illuminated by unpolarized light, scattered light will be partially polarized. For description of scattered light polarization state, the degree of linear polarization was introduced. For Rayleigh lightscattering, it can be described in following manner:
P=
2 I⊥ − I|| sin α = I⊥ + I|| 1 + cos2 α
(2)
When the particles are large enough to be comparable with the wavelength, the Rayleigh approximation cannot be used. By the way, scattering properties of spherical particles can be calculated analytically, and is called Mie scattering after Gustav Mie the first to publish the solution [7]. This fundamental theory has been used in great number of publications (see, e.g., [8–12] and references therein). There are numerous codes available to calculate Mie scattering [13,14]. We developed our own code, which allow making enough precise calculations of intensity and polarization of spher-
2
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
ical particles, and using it to studying the polarization behavior with radius and phase angle changing.
2. Quasi-rayleigh polarization leap Scattering by spherical particles has been studied well in a huge variety of different works. Usually the dependence of intensity and polarization of scattered light on phase angle was investigated. We decided to go a little different way and construct the dependences of the polarization of light scattered by small monodisperse spheres on the size parameter for different phase angles. Fig. 1a and b show the degree of linear polarization of light, scattered by monodisperse spheres, as a function of size parameter X. Fig. 1a calculated at refractive index m = 1.431+ 0i (weakly absorbing particle). Fig. 1b calculated at refractive index m = 1.294+ 0.1i (strongly absorbing particle). As can be seen, at small size parameters (X < 1) the degree of linear polarization is almost constant, corresponding to polarization of Rayleigh lightscattering (Eq. (2)). But at higher size parameters P is increasing, and then undergoes a quick leap to negative values. The area of size parameters corresponding to this leap is very narrow and depends both on refractive index and on phase angle. We suggest calling this phenomenon “quasi-rayleigh polarization leap”. This feature corresponds to the position of the minimum interference of light which is polarized perpendicular to the scattering plane. To calculate the polarization with an accuracy of 1% in the range of size parameters from 0 to 4.5, it is necessary to take at least 6 terms in the Mie series for weakly absorbing particles and 7 terms for strongly absorbing particles. Thus, approximate equations based on the consideration of only the first few terms of the Mie series [6] are not applicable to the quasi-rayleigh polarization leap study. Thereby, a tempting idea arises: to use this leap as a method of remote sensing of remote objects, in particular, determining the radius of spherical scattering particles. As well known, photometry is used much more often than polarimetry. Therefore, one should first find out: is there a similar leap in the case of scattered light intensity? Fig. 1c and d correspond to the same parameters, but show the intensity of scattered light. As one can see, intensity do not show fairly narrow and strong leap. So, this feature is manifested exclusively in degree of linear polarization. It is further interesting to find out in which area of the phase angles, size parameters and refractive indices one can observe a quasi-rayleigh polarization leap? To answer on this question, we build maps of the degree of linear polarization for different values of refractive indices (see Fig. 2). The X axis shows the phase angle values, and the Y axis shows the size parameter values. The degree of linear polarization corresponds to different shades of gray, the darker, the less polarization. The left column corresponds to the refractive indices m = 1.05+0i, m = 1.3 + +0i and m = 1.5 + +0i (weakly absorbing particles), the right one - to the refractive indices m = 1.05+0.1i, m = 1.3 + +0.1i and m = 1.5 + +0.1i (strongly absorbing particles). These maps clearly show that quasi-rayleigh polarization leap exists in wide range of phase angles and refractive indices. Moreover, the very interesting thing is the mutual increasing both size parameters and phase angles, at which quasi-rayleigh polarization leap is observed. In order to characterize the position of the quasi-rayleigh polarization leap, it is required to introduce a precise definition of the parameters that define it. We propose to consider the position of the quasi-rayleigh polarization leap as such values of the phase angle α min and size parameter Xmin at which the derivative of the
Fig. 1. Degree of linear polarization (a, b) and intensity (c, d) of light, scattered by sphere, as a function of sphere’s size parameter at different phase angles. Fig. 1a and c corresponds to weakly absorbing scatterer (refractive index m = 1.431+ 0i), Fig. 1b and d corresponds to strongly absorbing scatterer (refractive index m = 1.294+ 0.1i). Note that quasi-rayleigh leap exists in polarization and is absent in intensity.
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
3
Fig. 2. Maps of degree of linear polarization distributed over phase angles and size parameters. White color corresponds to high polarization values, black – low ones. Scales are on the right side of figures. Figures on left panel corresponds to weakly absorbing scatterer (k = 0), figures on right panel corresponds to strongly absorbing scatterer (k = 0.1). Figures on top row correspond to real part of refractive index n = 1.05, middle row - n = 1.3, bottom row - n = 1.50.
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function of the degree of linear polarization on the size parameter dP reaches the minimum value. dX Finding a derivative of a rapidly changing function, such as the degree of linear polarization near the quasi-rayleigh polarization leap, is not a quite simple task. For this, we had to develop our own program code to accurately calculate the degree of linear polarization of spherical particles. Derivatives were found numerically using the Ridders’ method [15]. dP Fig. 3 shows maps of dX values at the same parameters as dP Fig. 2. Here the brighter colors correspond to greater dX values. Unfortunately, grayscale picture cannot clearly shows the zero value, that is why this figure contain lines, each of them corredP sponds to condition dX = 0. It can be seen, that quasi-rayleigh polarization leap is most prominent at phase angles 40–50° for all refractive indices, except refractive index with relatively small real part n = 1.05. By the way, polarization leaps could be seen not only in quasirayleigh size parameters area. How can we be sure that the observed polarization leap is really quasi- rayleigh? This problem can be solved by enough simple way – we should make size parameter smaller, and in the case of quasi-rayleigh polarization leap - polarization should aim to a constant value determined by the Eq. (2).
3. Main features of quasi-rayleigh polarization leap As has already been mentioned in Section 2, quasi-rayleigh polarization leap exists in wide range of phase angles and refractive indices. Let us consider the refractive index dependences of main features of this phenomenon. dP Fig. 4a shows the dependences of minimal value of dX on the real part of refractive index n, at zero imaginary part. The deepest quasi-rayleigh polarization leap turns out to be in the region of low refractive index values, and the depth of the quasi-rayleigh polarization leap non-monotonously depends on the phase angle - for each value of the refraction index there is a phase angle at which the depth is maximal. However, in the phase angle range from 40° to 60°, the leap appears mostly clear. Fig. 4b demonstrates size parameter Xmin , at which quasirayleigh polarization leap exist at different phase angle. Difference in Xmin for various phase angles is maximal at small refractive indices and becomes negligible at large ones. Note, that large phase angles correspond to large values of Xmin , therefore for high values of observation phase angles equipment operating at shorter wavelengths is required. Finally, Fig. 4c shows the width of quasi-rayleigh polarization leap. Here we define the width of the leap as the difference in dimensional parameters at half the depth (Fig. 4a). Curves at phase angles <40° are not shown because they are very close to the lower curves. Fig. 5a,b and c show the same dependencies, but for refractive index imaginary part k = 0.1. Principal differences can be seen on Fig. 5a only, where, at small values of refractive index real part, quasi-rayleigh polarization leap has a relatively small depth. Moreover, depth at smaller phase angles was not shown, because the behavior of quasi-rayleigh polarization leap at the lower phase angles becomes non-monotonic, and the leap can be manifested very weakly at some combinations the phase angle-refractive index. Therefore, it is recommended to be very careful, when you try to detect quasi-rayleigh polarization leap for strongly absorbing particles at phase angles smaller than 40° In all other cases the curves behavior is quantitatively the same. We can conclude, that at low phase angles <40° for weakly absorbing particles, quasi-rayleigh polarization leap has a sufficiently small depth and width; for strongly absorbing particles, attempts to detect quasi-rayleigh polarization leap in this region of phase angles are also not recommended.
At phase angles >60° quasi-rayleigh polarization leap becomes very wide, but its deep tends to decrease, which makes it difficult both to detect it and to reveal the position of the minimum of the derivative. Therefore, observations in this phase angles area should be carried out very carefully and thoroughly. Lastly, 40°−60° phase angles region is the most promising in the aspect of confident detection and research of quasi-rayleigh polarization leap. Knowing the size parameter, phase angle and wavelength at which the quasi-rayleigh polarization leap is observed, as well as the refractive index of the scattering sphere at a given wavelength, one can determine the radius of the monodisperse spherical scatterer. This approach is potentially applicable to a vast array of celestial bodies. Further we will try to illustrate our approach, applying it to the polystyrene beads in Earth’s atmosphere. 4. Possible application of quasi-rayleigh polarization leap to polystyrene beads in Earth’s atmosphere Of course, fact of quasi-rayleigh polarization leap was noted by researchers earlier, for example, in some figures or contour maps in [6]. But in this paper we studied its properties and features in details, and also indicated the method of its use for specifying the polystyrene particle radius in the Earth’s atmosphere. A lot of aerosol particles there are in the atmosphere of Earth. They have a significant influence on human life, ranging from lightscattering observations to the climate changes processes [16,17] owing to the scattering and absorption of the electromagnetic waves. Aerosol particles can scatter light in different manners, depending on their size and chemical composition. That is why the studying of atmospheric aerosols is a very important. A prominent role among atmospheric aerosols is played by polystyrene beads. Polystyrene is widely used as food containers and packaging and is generated from both industrial and municipal polymers sources. It has becoming a major environmental concern due to large quantities disposed to landfills and its non-biodegradable in nature [18]. Polystyrene beads, due to their features, such as the spherical shape and monodispersity, are often used by researchers as a test aerosol. Polystyrene beads are commonly used for characterization and calibration of instruments, which remotely sensing the atmosphere aerosol’s particles [19]. These challenges require a careful characterization of test aerosol particles. There is a significant uncertainty in determining the particle size of polystyrene in the atmosphere. Particle sizes estimations vary from R = 1.227 μm [20] up to R = 1.299 μm [21]. That is why refining the particle size of polystyrene seems to be a worthwhile task. Meanwhile, the variations of polystyrene particles size seems to be very small (±0.0 08 μm [22], ±0.0 09 μm [20]). So such particles could be considered as monodisperese. Measurements of polystyrene beads refractive index were carried out by Jones and colleagues [22]. They measured the scattering spectra of polystyrene beads samples in the range 480–700 nm with a resolution of 0.028 nm. They found, that polystyrene beads have the dispersion of refractive index described by equation:
n = 1.5718 +
8412
λ2
+
2.35 × 108
λ4
+ 0.0 0 03i,
(3)
where n is complex refractive index and λ is the wavelength of incident light. Eq. (1) is valid for λ expressed in nanometers and in spectral range 480–700 nm. But the measurements of polystyrene refractive index in infrared area, carried out by Liang et al. [23], show that the refractive index of polystyrene does not undergo significant changes up to a wavelength of 5.5 μm. So we can suggest that the dependence (3) is valid up to this wavelength.
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
Fig. 3. The same as Fig. 2, but for derivation of linear polarization degree over size parameter
Based on this suggestion, we calculate the radius of scattering particles, defined by position of quasi-rayleigh polarization leap both over phase angle α min and over wavelength λmin (Fig. 6). As can be seen, dependencies are approximately linear due to small changes of refractive index. We provide an approximation of the
dP dX
. Lines correspond to constant polarization
5
dP dX
= 0.
function of two variables, which describe the quasi-rayleigh polarization leap of polystyrene beads:
RPB (αmin , λmin ) = −0.04302 + 0.27143λmin + 0.0010555αmin (4)
6
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
Fig. 4. Dependencies of the quasi-rayleigh polarization leap parameters (a – depth dP ); b – position Xmin ; c – half-width Xmin ) on real part of refractive index min( dX at zero imaginary part (k = 0).
Fig. 5. The same as Fig. 4, but for refractive index imaginary part k = 0.1.
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
7
mosphere of Earth, based on two parameters — the wavelength and the phase angle at which quasi-rayleigh polarization leap is observed. It should be noted that there may be many leaps of polarization, especially in the case of scattering by spherical particles, at different phase angles and wavelengths. Therefore, to find out whether this leap is precisely quasi-rayleigh, one should proceed as designated in Section 2. It is necessary to increase the wavelength at which observations are made (which corresponds to a decrease in the size parameter), and the polarization very soon should strive to a constant value depending on phase angle as Eq. (2). Declaration of Competing Interest None. CRediT authorship contribution statement Dmitry Petrov: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision, Project administration. Elena Zhuzhulina: Funding acquisition. References
Fig. 6. Wavelength dependence of polystyrene refractive index (upper panel) and radius of scattering particles of polystyrene beads, defined by position of quasirayleigh polarization leap both over phase angle α min and over wavelength λmin (lower panel).
Eq. (4) is valid in phase angle area α min = 30°…70° and in wavelength area λmin = 4.0…5.4 μm. Note, that α min should be expressed in degrees and λmin in micrometers. 5. Concluding remarks In this paper, the phenomenon of a quick change of the degree of linear polarization of monodisperse spherical particles has been studied. This feature called «quasi-rayleigh polarization leap» appears with increasing of scattering particle radius, when it becomes in several times higher than wavelength. Regularities that determine the position of quasi-rayleigh polarization leap depending on the particle radius, refractive index, phase angle, and wavelength of observations have been established. Of course, fact of quasi-rayleigh polarization leap was noted by researchers earlier [6], but we studied its properties and features, and also indicated the method of its use for specifying the particle radius of the polystyrene beads in Earth’s atmosphere. On this basis, it was suggested to use quasi-rayleigh polarization leap for remote sensing the radius of distant spherical objects. A simple empirical formula (Eq. (4)) has been obtained that allows us to calculate the radius of polystyrene bead particles in the at-
[1] Strutt JW. XV. On the light from the sky, its polarization and colour. London Edinburgh Dublin Philos Mag J Sci 1871;41(271):107–20. doi:10.1080/ 14786447108640452. [2] Strutt JW. XXXVI. On the light from the sky, its polarization and colour. London Edinburgh Dublin Philos Mag J Sci 1871;41(273):274–9. doi:10.1080/ 14786447108640479. [3] Strutt JW. LVIII. On the scattering of light by small particles. London Edinburgh Dublin Philos Mag J Sci 1871;41(275):447–54. doi:10.1080/ 14786447108640507. [4] Rayleigh Lord. X. On the electromagnetic theory of light. London Edinburgh Dublin Philos Mag J Sci 1881;12(73):81–101. doi:10.1080/14786448108627074. [5] Rayleigh Lord. XXXIV. On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky. London Edinburgh Dublin Philos Mag J Sci 1899;47(287):375–84. doi:10.1080/14786449908621276. [6] Van de Hulst HC. Light scattering by small particles. Dover 1981 10. doi:10. 1063/1.3060205. [7] Mie G. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann Phys 1908;330(3):377–445. doi:10.10 02/andp.1908330 0302. [8] Kahnert FM. Numerical methods in electromagnetic scattering theory. J Quant Spectrosc Radiative Transfer 2003:775–824 79–80. doi:10.1016/ S0 022-4073(02)0 0321-7. [9] Mishchenko MI, Lacis AA. Morphology-dependent resonances of nearly spherical particles in random orientation. Appl Opt 2003;42:5551–6. doi:10.1364/AO. 42.005551. [10] Cai W, Zhao Y, Ma L. Direct recursion of the ratio of Bessel functions with applications to MIE scattering calculations. J Quant Spectrosc Radiative Transfer 2008;109:2673–8. doi:10.1016/j.jqsrt.2008.07.009. [11] Horvath H. Gustav MIE and the scattering and absorption of light by particles: historic developments and basics. J Quant Spectrosc Radiative Transfer 2009;110:787–99. doi:10.1016/j.jqsrt.2009.02.022. [12] Spurr R, Wang J, Zeng J, Mishchenko MI. Linearized T-matrix and MIE scattering computations. J Quant Spectrosc Radiative Transfer 2012;113:425–39. doi:10.1016/j.jqsrt.2011.11.014. [13] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. Cambridge: Cambridge University Press; 2002. [14] Mishchenko MI, Dlugach JM, Yanovitskij EG, Zakharova NT. Bidirectional reflectance of flat, optically thick particulate laters: an efficient radiative transfer solution and applications to snow and soil surfaces. J Quant Spectrosc Radiative Transfer 1999;63:409–32. [15] Ridders CJF. Adv Eng Software 1982;4(2):75–6. [16] Cao J. The importance of aerosols in the earth system: science and engineering perspectives. Aerosol Sci Eng 2017;1(1):1–6. doi:10.1007/s41810- 017- 0005- 1. [17] Tariq S, ul-Haq Z. Investigating the aerosol optical depth and angstrom exponent and their relationships with meteorological parameters over Lahore in Pakistan. In: Proceedings of the national academy of sciences, india section a: physical sciences; 2019. p. 1–7. doi:10.10 07/s40 010- 018- 0575- 6. [18] Ewadh HM, Basri NA. Effectiveness of polystyrene beads as aggregate replacement material to recycle solid waste: a study on workability and absorption results of concrete. Int J Sci Eng Res 2012;3(8):26–9. [19] Miles REH, Rudic´ S, Orr-Ewing AJ, Reid JP. Influence of uncertainties in the diameter and refractive index of calibration polystyrene beads on the retrieval
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of aerosol optical properties using cavity ring down spectroscopy. J Phys Chem A 2010;114:7077–84. doi:10.1021/jp103246t. [20] Bateman JB, Weneck EJ, Eshler DC. Determination of particle size and concentration from spectrophotometric transmission. J Colloid Sci 1959;Volume 14(Issue 3):308–29. doi:10.1016/0 095-8522(59)90 055-8. [21] Ma X, Lu JW, Brock RS, Jacobs KM, Yang P, Hu X-H. Determination of complex refractive index of polystyrene microspheres from 370 to 1610nm. Phys Med Biol 2003;V. 48:4165–72. doi:10.1088/0031-9155/48/24/013.
[22] Jones SH, King MD, Ward AD. Determining the unique refractive index properties of solid polystyrene aerosol using broadband mie scattering from optically trapped beads. Physical Chemistry Chemical Physics 2013;15:20735–41. doi:10.1039/C3CP53498G. [23] Jianfeng L, Min T, Wenyuan L, Zhi Q, Gangqiang C, Wenyu P. Optical constant determination of cross-linked polystyrene in the infrared. In: Proc. SPIE 10623, 2017 international conference on optical instruments and technology: IRMMW-THz technologies and applications; 2018. 106230B doi:10.1117/12.2292416.
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Quasi-rayleigh polarization leap of monodisperse spherical particle as a tool to detect particle radius Dmitry Petrov∗, Elena Zhuzhulina Crimean Astrophysical Observatory (CrAO RAS), Nauchnyj, Crimea 298409, Russia
a r t i c l e
i n f o
Article history: Received 24 May 2019 Revised 16 December 2019 Accepted 16 December 2019 Available online 17 December 2019 Keywords: Mie theory Rayleigh scattering Polarization Polystyrene beads
a b s t r a c t When the size of scattering particle is increased to be comparable with wavelength, Rayleigh approximation becomes not valid. In polarization scattering pattern this quasi-rayleigh behavior manifests itself as a sharp drop of the degree of linear polarization to the negative values. In this paper, we studied the properties and features of this quasi-rayleigh polarization leap for monodisperse spherical scatterers. The main regularities that determine the wavelength position of the quasi-rayleigh polarization leap are established depending on the phase angle, refractive index and size parameter of the scatterers. A method is suggested for determining the radius of spherical scattering particles of the polystyrene beads in Earth’s atmosphere. A simple interpolation formula is given, which allows, based on observational data, such as phase angle and the wavelength at which the quasi-rayleigh polarization leap is observed, to calculate the particle radius. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Rayleigh lightscattering means the scattering of light by enough small objects. Main properties of such kind of scattering were originally described by Rayleigh [1–5] to explain the color and polarization of the light from the sky. More generally, it describes the scattering of light by particles much smaller than the wavelength of the light, and smaller than the wavelength of the light divided by the real part of the complex refractive index m = n + i·k. As well known, the scattering properties of any object depend on the ratio between the size and wavelength of light. For this reason, the scattering particle is characterized by a size paramer ter X = 2π λ , where λ is the wavelength of the incident light, and r in the case of spherical scatterer is the radius of sphere [6]. For Rayleigh scattering, the following condition should be complied: X|n| <<1. When such a small particle is illuminated by electromagnetic waves, every part of particle experiences the electromagnetic field simultaneously. This field results in the charges separation, so the scattering particle acquires a dipole moment, which changes in consistent with the external electromagnetic field of incident light. The particle then radiates scattered light as a result of the accelerating movement of charges. An oscillating dipole produces an oscillating electromagnetic scattered field, which is proportional to the squared frequency of the oscillation. Since the intensity of light is
∗
Corresponding author. E-mail address: [email protected] (D. Petrov).
https://doi.org/10.1016/j.jqsrt.2019.106806 0022-4073/© 2019 Elsevier Ltd. All rights reserved.
proportional to the square of the amplitude of the electric field, the resulting intensity of scattering depends on wavelength as frequency to the forth power, i.e. I ∼ λ−4 . Moreover, the intensity of scattered light depends on the phase angle α , i.e. on the angle between the direction lightsource - scatterer and scatterer - observer. The phase function of Rayleigh scattering depends on the phase angle as:
I∼
1 + cos2 α
λ4
.
(1)
Even when scattering particle is illuminated by unpolarized light, scattered light will be partially polarized. For description of scattered light polarization state, the degree of linear polarization was introduced. For Rayleigh lightscattering, it can be described in following manner:
P=
2 I⊥ − I|| sin α = I⊥ + I|| 1 + cos2 α
(2)
When the particles are large enough to be comparable with the wavelength, the Rayleigh approximation cannot be used. By the way, scattering properties of spherical particles can be calculated analytically, and is called Mie scattering after Gustav Mie the first to publish the solution [7]. This fundamental theory has been used in great number of publications (see, e.g., [8–12] and references therein). There are numerous codes available to calculate Mie scattering [13,14]. We developed our own code, which allow making enough precise calculations of intensity and polarization of spher-
2
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
ical particles, and using it to studying the polarization behavior with radius and phase angle changing.
2. Quasi-rayleigh polarization leap Scattering by spherical particles has been studied well in a huge variety of different works. Usually the dependence of intensity and polarization of scattered light on phase angle was investigated. We decided to go a little different way and construct the dependences of the polarization of light scattered by small monodisperse spheres on the size parameter for different phase angles. Fig. 1a and b show the degree of linear polarization of light, scattered by monodisperse spheres, as a function of size parameter X. Fig. 1a calculated at refractive index m = 1.431+ 0i (weakly absorbing particle). Fig. 1b calculated at refractive index m = 1.294+ 0.1i (strongly absorbing particle). As can be seen, at small size parameters (X < 1) the degree of linear polarization is almost constant, corresponding to polarization of Rayleigh lightscattering (Eq. (2)). But at higher size parameters P is increasing, and then undergoes a quick leap to negative values. The area of size parameters corresponding to this leap is very narrow and depends both on refractive index and on phase angle. We suggest calling this phenomenon “quasi-rayleigh polarization leap”. This feature corresponds to the position of the minimum interference of light which is polarized perpendicular to the scattering plane. To calculate the polarization with an accuracy of 1% in the range of size parameters from 0 to 4.5, it is necessary to take at least 6 terms in the Mie series for weakly absorbing particles and 7 terms for strongly absorbing particles. Thus, approximate equations based on the consideration of only the first few terms of the Mie series [6] are not applicable to the quasi-rayleigh polarization leap study. Thereby, a tempting idea arises: to use this leap as a method of remote sensing of remote objects, in particular, determining the radius of spherical scattering particles. As well known, photometry is used much more often than polarimetry. Therefore, one should first find out: is there a similar leap in the case of scattered light intensity? Fig. 1c and d correspond to the same parameters, but show the intensity of scattered light. As one can see, intensity do not show fairly narrow and strong leap. So, this feature is manifested exclusively in degree of linear polarization. It is further interesting to find out in which area of the phase angles, size parameters and refractive indices one can observe a quasi-rayleigh polarization leap? To answer on this question, we build maps of the degree of linear polarization for different values of refractive indices (see Fig. 2). The X axis shows the phase angle values, and the Y axis shows the size parameter values. The degree of linear polarization corresponds to different shades of gray, the darker, the less polarization. The left column corresponds to the refractive indices m = 1.05+0i, m = 1.3 + +0i and m = 1.5 + +0i (weakly absorbing particles), the right one - to the refractive indices m = 1.05+0.1i, m = 1.3 + +0.1i and m = 1.5 + +0.1i (strongly absorbing particles). These maps clearly show that quasi-rayleigh polarization leap exists in wide range of phase angles and refractive indices. Moreover, the very interesting thing is the mutual increasing both size parameters and phase angles, at which quasi-rayleigh polarization leap is observed. In order to characterize the position of the quasi-rayleigh polarization leap, it is required to introduce a precise definition of the parameters that define it. We propose to consider the position of the quasi-rayleigh polarization leap as such values of the phase angle α min and size parameter Xmin at which the derivative of the
Fig. 1. Degree of linear polarization (a, b) and intensity (c, d) of light, scattered by sphere, as a function of sphere’s size parameter at different phase angles. Fig. 1a and c corresponds to weakly absorbing scatterer (refractive index m = 1.431+ 0i), Fig. 1b and d corresponds to strongly absorbing scatterer (refractive index m = 1.294+ 0.1i). Note that quasi-rayleigh leap exists in polarization and is absent in intensity.
D. Petrov and E. Zhuzhulina / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106806
3
Fig. 2. Maps of degree of linear polarization distributed over phase angles and size parameters. White color corresponds to high polarization values, black – low ones. Scales are on the right side of figures. Figures on left panel corresponds to weakly absorbing scatterer (k = 0), figures on right panel corresponds to strongly absorbing scatterer (k = 0.1). Figures on top row correspond to real part of refractive index n = 1.05, middle row - n = 1.3, bottom row - n = 1.50.
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function of the degree of linear polarization on the size parameter dP reaches the minimum value. dX Finding a derivative of a rapidly changing function, such as the degree of linear polarization near the quasi-rayleigh polarization leap, is not a quite simple task. For this, we had to develop our own program code to accurately calculate the degree of linear polarization of spherical particles. Derivatives were found numerically using the Ridders’ method [15]. dP Fig. 3 shows maps of dX values at the same parameters as dP Fig. 2. Here the brighter colors correspond to greater dX values. Unfortunately, grayscale picture cannot clearly shows the zero value, that is why this figure contain lines, each of them corredP sponds to condition dX = 0. It can be seen, that quasi-rayleigh polarization leap is most prominent at phase angles 40–50° for all refractive indices, except refractive index with relatively small real part n = 1.05. By the way, polarization leaps could be seen not only in quasirayleigh size parameters area. How can we be sure that the observed polarization leap is really quasi- rayleigh? This problem can be solved by enough simple way – we should make size parameter smaller, and in the case of quasi-rayleigh polarization leap - polarization should aim to a constant value determined by the Eq. (2).
3. Main features of quasi-rayleigh polarization leap As has already been mentioned in Section 2, quasi-rayleigh polarization leap exists in wide range of phase angles and refractive indices. Let us consider the refractive index dependences of main features of this phenomenon. dP Fig. 4a shows the dependences of minimal value of dX on the real part of refractive index n, at zero imaginary part. The deepest quasi-rayleigh polarization leap turns out to be in the region of low refractive index values, and the depth of the quasi-rayleigh polarization leap non-monotonously depends on the phase angle - for each value of the refraction index there is a phase angle at which the depth is maximal. However, in the phase angle range from 40° to 60°, the leap appears mostly clear. Fig. 4b demonstrates size parameter Xmin , at which quasirayleigh polarization leap exist at different phase angle. Difference in Xmin for various phase angles is maximal at small refractive indices and becomes negligible at large ones. Note, that large phase angles correspond to large values of Xmin , therefore for high values of observation phase angles equipment operating at shorter wavelengths is required. Finally, Fig. 4c shows the width of quasi-rayleigh polarization leap. Here we define the width of the leap as the difference in dimensional parameters at half the depth (Fig. 4a). Curves at phase angles <40° are not shown because they are very close to the lower curves. Fig. 5a,b and c show the same dependencies, but for refractive index imaginary part k = 0.1. Principal differences can be seen on Fig. 5a only, where, at small values of refractive index real part, quasi-rayleigh polarization leap has a relatively small depth. Moreover, depth at smaller phase angles was not shown, because the behavior of quasi-rayleigh polarization leap at the lower phase angles becomes non-monotonic, and the leap can be manifested very weakly at some combinations the phase angle-refractive index. Therefore, it is recommended to be very careful, when you try to detect quasi-rayleigh polarization leap for strongly absorbing particles at phase angles smaller than 40° In all other cases the curves behavior is quantitatively the same. We can conclude, that at low phase angles <40° for weakly absorbing particles, quasi-rayleigh polarization leap has a sufficiently small depth and width; for strongly absorbing particles, attempts to detect quasi-rayleigh polarization leap in this region of phase angles are also not recommended.
At phase angles >60° quasi-rayleigh polarization leap becomes very wide, but its deep tends to decrease, which makes it difficult both to detect it and to reveal the position of the minimum of the derivative. Therefore, observations in this phase angles area should be carried out very carefully and thoroughly. Lastly, 40°−60° phase angles region is the most promising in the aspect of confident detection and research of quasi-rayleigh polarization leap. Knowing the size parameter, phase angle and wavelength at which the quasi-rayleigh polarization leap is observed, as well as the refractive index of the scattering sphere at a given wavelength, one can determine the radius of the monodisperse spherical scatterer. This approach is potentially applicable to a vast array of celestial bodies. Further we will try to illustrate our approach, applying it to the polystyrene beads in Earth’s atmosphere. 4. Possible application of quasi-rayleigh polarization leap to polystyrene beads in Earth’s atmosphere Of course, fact of quasi-rayleigh polarization leap was noted by researchers earlier, for example, in some figures or contour maps in [6]. But in this paper we studied its properties and features in details, and also indicated the method of its use for specifying the polystyrene particle radius in the Earth’s atmosphere. A lot of aerosol particles there are in the atmosphere of Earth. They have a significant influence on human life, ranging from lightscattering observations to the climate changes processes [16,17] owing to the scattering and absorption of the electromagnetic waves. Aerosol particles can scatter light in different manners, depending on their size and chemical composition. That is why the studying of atmospheric aerosols is a very important. A prominent role among atmospheric aerosols is played by polystyrene beads. Polystyrene is widely used as food containers and packaging and is generated from both industrial and municipal polymers sources. It has becoming a major environmental concern due to large quantities disposed to landfills and its non-biodegradable in nature [18]. Polystyrene beads, due to their features, such as the spherical shape and monodispersity, are often used by researchers as a test aerosol. Polystyrene beads are commonly used for characterization and calibration of instruments, which remotely sensing the atmosphere aerosol’s particles [19]. These challenges require a careful characterization of test aerosol particles. There is a significant uncertainty in determining the particle size of polystyrene in the atmosphere. Particle sizes estimations vary from R = 1.227 μm [20] up to R = 1.299 μm [21]. That is why refining the particle size of polystyrene seems to be a worthwhile task. Meanwhile, the variations of polystyrene particles size seems to be very small (±0.0 08 μm [22], ±0.0 09 μm [20]). So such particles could be considered as monodisperese. Measurements of polystyrene beads refractive index were carried out by Jones and colleagues [22]. They measured the scattering spectra of polystyrene beads samples in the range 480–700 nm with a resolution of 0.028 nm. They found, that polystyrene beads have the dispersion of refractive index described by equation:
n = 1.5718 +
8412
λ2
+
2.35 × 108
λ4
+ 0.0 0 03i,
(3)
where n is complex refractive index and λ is the wavelength of incident light. Eq. (1) is valid for λ expressed in nanometers and in spectral range 480–700 nm. But the measurements of polystyrene refractive index in infrared area, carried out by Liang et al. [23], show that the refractive index of polystyrene does not undergo significant changes up to a wavelength of 5.5 μm. So we can suggest that the dependence (3) is valid up to this wavelength.
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Fig. 3. The same as Fig. 2, but for derivation of linear polarization degree over size parameter
Based on this suggestion, we calculate the radius of scattering particles, defined by position of quasi-rayleigh polarization leap both over phase angle α min and over wavelength λmin (Fig. 6). As can be seen, dependencies are approximately linear due to small changes of refractive index. We provide an approximation of the
dP dX
. Lines correspond to constant polarization
5
dP dX
= 0.
function of two variables, which describe the quasi-rayleigh polarization leap of polystyrene beads:
RPB (αmin , λmin ) = −0.04302 + 0.27143λmin + 0.0010555αmin (4)
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Fig. 4. Dependencies of the quasi-rayleigh polarization leap parameters (a – depth dP ); b – position Xmin ; c – half-width Xmin ) on real part of refractive index min( dX at zero imaginary part (k = 0).
Fig. 5. The same as Fig. 4, but for refractive index imaginary part k = 0.1.
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mosphere of Earth, based on two parameters — the wavelength and the phase angle at which quasi-rayleigh polarization leap is observed. It should be noted that there may be many leaps of polarization, especially in the case of scattering by spherical particles, at different phase angles and wavelengths. Therefore, to find out whether this leap is precisely quasi-rayleigh, one should proceed as designated in Section 2. It is necessary to increase the wavelength at which observations are made (which corresponds to a decrease in the size parameter), and the polarization very soon should strive to a constant value depending on phase angle as Eq. (2). Declaration of Competing Interest None. CRediT authorship contribution statement Dmitry Petrov: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision, Project administration. Elena Zhuzhulina: Funding acquisition. References
Fig. 6. Wavelength dependence of polystyrene refractive index (upper panel) and radius of scattering particles of polystyrene beads, defined by position of quasirayleigh polarization leap both over phase angle α min and over wavelength λmin (lower panel).
Eq. (4) is valid in phase angle area α min = 30°…70° and in wavelength area λmin = 4.0…5.4 μm. Note, that α min should be expressed in degrees and λmin in micrometers. 5. Concluding remarks In this paper, the phenomenon of a quick change of the degree of linear polarization of monodisperse spherical particles has been studied. This feature called «quasi-rayleigh polarization leap» appears with increasing of scattering particle radius, when it becomes in several times higher than wavelength. Regularities that determine the position of quasi-rayleigh polarization leap depending on the particle radius, refractive index, phase angle, and wavelength of observations have been established. Of course, fact of quasi-rayleigh polarization leap was noted by researchers earlier [6], but we studied its properties and features, and also indicated the method of its use for specifying the particle radius of the polystyrene beads in Earth’s atmosphere. On this basis, it was suggested to use quasi-rayleigh polarization leap for remote sensing the radius of distant spherical objects. A simple empirical formula (Eq. (4)) has been obtained that allows us to calculate the radius of polystyrene bead particles in the at-
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