Single-scattering Phase Function Research Articles

Three Years of ACB Phase Function Observations from the Mars Science Laboratory: Interannual and Diurnal Variability and Constraints on Ice Crystal Habit

We examine 3 yr of phase-function observations of water-ice clouds taken during the Aphelion Cloud Belt season by the Mars Science Laboratory (MSL). We derive lower-bound single-scattering phase functions for Mars years (MYs) 34, 35, and 36, over a range of scattering angles from 45° to 155°, expanding on the MY 34 phase function previously derived from MSL observations using the same method. We also modify the procedure used for MY 34 to make use of cloud opacity values derived from other MSL observations, often taken in conjunction with the phase-function observations. From these, we see little variability, both interannually and diurnally in the phase function at Gale Crater. We use our derived phase functions to attempt to constrain a dominant ice-crystal geometry by fitting a two-term Henyey–Greenstein function. In comparing to HG functions of Martian dust and modeled water-ice crystals, we see agreement especially with droxtal water-ice crystals, dust at Gale crater, and irregular volcanic glasses. This could be indicative of crystals composed of some irregular shape.

Quantifying the coherent backscatter enhancement of non-spherical particles with discrete dipole approximation.

The prevailing backscattering peak associated with the scattering phase function of large non-absorptive particles can be interpreted with the coherent backscatter enhancement (CBE) theory, but has not been explicitly quantified with numerical simulations based on solving Maxwell's equations. In this paper, representative numerical simulations performed with the discrete-dipole-approximation (DDA) model are used to quantify the effect of CBE on the single-scattering phase function. For each scattering case, the particle volume was divided into multiple thin slices parallel to the incident beam. The dipole polarizations in the j'th slice in response to the incident field of the i'th slice were computed, and then the corresponding contribution to the scattering phase function was calculated. Interference between conjugate terms representing reversible wave paths is constructive at the backscattering direction, which corresponds to the CBE. Subsequently, the contribution of CBE to the scattering phase function was quantified by comparing the electric fields calculated with and without the interference between conjugate terms. Results from these numerical simulations are consistent with conclusions obtained from the CBE theory. The simulations also quantitatively explain why it is difficult to identify a CBE-induced backscattering peak for the phase function of small particles and strong-absorptive particles.

Cirrus-cloud conditioned ratio of multiple-to-single backscatter atmospheric phase functions

The present work is related to LIDAR remote sensing of the atmosphere in the presence of cirrus clouds (CC) along the LIDAR line of sight (LOS), when the multiple-backscatter component of the LIDAR return may be essential. E. Eloranta has found analytical expressions of the multiple-backscatter components of different orders as containing the corresponding ratios of the multiple-to-single backscatter phase functions of the backward scattering medium. As a main purpose of this work, we estimated the above-mentioned ratios in cases of backscattering from air and CC, using accurate approximations of the single-scattering phase functions obtained by U. Wandinger. The results obtained show that the multiple-to-single backscatter phase function ratios for air are equal to unity, while for CCs they are quite near unity.

Effect of ground-based environmental conditions on the level of dangerous ultraviolet solar radiation

An analytical model is suggested to determine the contribution of solar radiation scattered by dispersed layers of weakly absorbing highly-porous media on the surface of the earth to the local UV-index, which characterizes harmful ultraviolet irradiation. The transport approximation for the single-scattering phase function and the two-flux method for approximate radiative transfer calculations were used to calculate an additional UV irradiance due to multiple scattering in the layer of a substrate medium. Three important environment conditions are considered in the case problems: the snowpack in a mountain ski resort of Provo (Utah), the salt layer in the Great Salt Lake desert, and the layers of sand typical of sand resorts and beaches. The optical properties of the media were calculated using the data for spectral optical constants of substances in the near-ultraviolet and analytical geometrical optics solution for large spherical particles of a weakly absorbing material. The resulting spectral single-scattering transport albedo of the medium was used to determine an additional harmful UV radiation. It was found that corrected values of the UV-index in all three cases are substantially higher than those calculated without account for the light reflected from a scattering medium on the earth surface. The model of the present paper can be used for different dispersed media in a wide range of physical parameters.

Measurements of scattering and absorption properties of submillimeter bauxite and silica particles

Submillimeter solid particles have been considered as thermal storage media for concentrated solar power applications. Knowledge of the scattering and absorption properties of individual particles is crucial for modeling the radiative heat transfer of the particle bed. In this work, a laser scatterometer is used to measure the single-particle scattering properties at a wavelength of 635 nm by using two configurations: (1) a falling particle curtain and (2) a taped particle layer. Because a one-particle nominal thickness is formed with area fractions of 5−55% depending on the configuration, multiple scattering is minimized and hence the single scattering phase function, averaged over all illuminated particles, is directly measured. Bauxite-based ceramic particles that are strongly absorbing in the solar spectrum and silica particles that are nonabsorbing in the visible and near-infrared are investigated. The directional-hemispherical reflectance and transmittance of the taped particles are also measured to deduce the forward and backward scattering efficiency factors and the absorption efficiency factors. Only weak wavelength dependence is observed in the measured region from 380 nm to 1020 nm. The scattering phase functions of all bauxite-based particles with varying sizes and compositions are very similar and can be fitted to a Henyey−Greenstein phase function with an asymmetry factor g = −0.20. For the silica particles, forward scattering dominates and g = 0.45 yields the best fit. A Monte Carlo method is developed to model the particle scattering characteristics, and reasonable agreements between the modeling and experimental results are observed by introducing a specularity parameter.

Characteristics of Pluto’s Haze and Surface from an Analytic Radiative Transfer Model

Abstract Observations of Pluto from New Horizons have been combined with previous ground-based observations and fit to a radiative transfer model based on Chandrasekhar’s planetary problem and Hapke theory to simultaneously derive the physical properties of the dwarf planet’s surface and atmosphere. We derive the macroscopic roughness, single-scattering albedo, and directional scattering properties of the surface, and the single-scattering albedo, optical depth, and single-scattering phase function of Pluto’s haze. The haze particles are small, with best-fit sizes in the range of ∼0.41–1.14 μm. We find that Pluto’s haze is more similar to that of Titan, rich in organic compounds and highly forward scattering, than that of Triton. With organic compounds and a likely subsurface water ocean, Pluto may harbor sustainable habitable environments. Our model, which includes the coherent backscatter effect, fits the anomalously large opposition surge recently discovered on Pluto.

Dust particle size, shape and optical depth during the 2018/MY34 martian global dust storm retrieved by MSL Curiosity rover Navigation Cameras

Martian planet-encircling dust storms or global dust storms (GDS), resulting from the combined influence of local and regional storms, are uncommon aperiodic phenomena: with an average frequency of approximately one every 3–4 MY, they produce a substantial rise in the atmospheric dust loading that lasts from weeks to months and have a significant impact on the atmospheric properties, energy budget, and global circulation. During the 2018/MY34 global dust storm, initiated at LS = 185° (30–31 May 2018), an intensive atmospheric science campaign was carried out by the Mars Science Laboratory (MSL) rover to monitor the environmental parameters at Gale Crater. We contribute to previous studies with independent retrievals to constrain the dust opacity and characterise the aerosol particle properties, including: size, shape and single scattering phase function. An iterative radiative transfer retrieval procedure was implemented to determine the aerosol parameters that best fit the angular distribution of sky radiance at forward and backward scattering regions observed by MSL Navigation Cameras (Navcams) during the 2018/MY34 GDS. The MOPSMAP aerosol database and Double Henyey-Greenstein (DHG) analytical single scattering phase functions were used to model the Martian dust aerosol. Outcomes of this study show a steep rise in dust opacity from pre-storm levels of 1.2 up to τ > 9, correlated to particle size variations from 1 to 4 μm. DHG phase functions are characterised with an average asymmetry parameter of g = 0.60 ± 0.11 during the storm, diverging from values of around 0.71 ± 0.06 for the same period in previous MY. Best fitting simulations to backscatter observations for high-opacity periods were generated by a mixture of spheroids following a log-normal distribution of aspect ratios centred on 2.8 ± 0.9, in contrast to values of 1.8 in post-storm sols, thus pointing to more irregular particle shapes at the peak of the dust storm.

Application and evaluation of the small-angle approximation on forward radiative transfer model

The small-angle approximation is used in solving the radiative transfer equation when the single-scattering phase function has a strong forward peak. The original vector radiative transfer equation is decomposed into three equations: the forward, the regular, and the correction ones. The forward equation can be efficiently solved using the small-angle approximation. The solution of the regular equation is given by the adding-doubling method in this study. The correction equation including cross terms between the forward and the regular quantities is given and analyzed. The combined model associated with the forward and regular equations is further verified in the forward radiative transfer model, where the molecular absorptions with respect to the standard atmospheric profiles are taken into consideration using the line-by-line method. Great agreements are given between the combined model and the model using the straight adding-doubling method.

A Fast All-sky Radiation Model for Solar applications with Narrowband Irradiances on Tilted surfaces (FARMS-NIT): Part II. The cloudy-sky model

The Fast All-sky Radiation Model for Solar applications with Narrowband Irradiances on Tilted surfaces (FARMS-NIT) reported in Part I of this study is enhanced to include the requirements for cloudy-sky conditions. Surface radiances in 2002 narrow-wavelength bands from 0.28 to 4.0 μm are analytically computed by solving the radiative transfer equation for five independent photon paths accounting for clear-sky absorption, Rayleigh scattering, and cloud absorption and scattering. The Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS) is used to provide the optical thickness of the clear-sky atmosphere. Unlike Part I, which approximates the computation of aerosol scattering using the single-scattering phase function, the cloud transmittance and reflectance are efficiently retrieved from a comprehensive look-up table pre-computed by a 32-stream DIScrete Ordinates Radiative Transfer (DISORT) model for possible cloud conditions as well as solar and viewing geometries. A resolution analysis is performed to assess the optimal balance between the computational efficiency and accuracy in the development of the look-up table. Model simulations by DISORT and TMYSPEC are used to evaluate the performance of FARMS-NIT under cloudy-sky conditions. Compared to DISORT, FARMS-NIT yields 2–3% uncertainties on average, but it substantially reduces the computational time because of the independent computation of cloud properties and the implementation of the look-up table. In contrast to TMYSPEC, which uses successive steps to empirically compute plane-of-array (POA) irradiances and spectral irradiances, FARMS-NIT directly solves spectral radiances from the radiative transfer equation, which profoundly increases the accuracy in surface irradiances, especially over inclined photovoltaics (PV) panels.

Characterisation of Martian dust aerosol phase function from sky radiance measurements by MSL engineering cameras

Dust is the main driver of Mars' atmospheric variability. The determination of Martian dust aerosol properties is of high relevance for radiative modelling and calculating its weather forcing. In particular, the light scattering behaviour at intermediate and large scattering angles can provide valuable information regarding the airborne dust particle shape. The angular distribution of sky brightness observed by the Mars Science Laboratory engineering cameras (Navcam and Hazcam) is used here to characterise the atmospheric dust single scattering phase function and to constrain the shape of the particles. An iterative radiative transfer based retrieval method was implemented in order to determine the aerosol modelling parameters which best reproduce the observed sky radiance as a function of the scattering angle in the solar almucantar plane. The aerosol models considered in this study for retrieving dust radiative properties were an analytical three term Double Henyey-Greenstein (DHG) phase function, T-matrix calculations for cylindrical particles with different diameter-to-length (D/L) aspect ratios and experimental phase functions from laboratory measurements of several Martian dust analogue samples. Results of this study returned mean DHG phase function parameter values g1 = 0.889 ± 0.098, g2 = 0.094 ± 0.250, α = 0.743 ± 0.106; generating a phase function with an asymmetry parameter of g = 0.673 ± 0.081 (in line with Wolff et al., 2009). Although differences were observed during the low opacity aphelion season (lower forward scattering values, presence of a peak in the backward region) compared to the rest of the year, no clear evidences of seasonal behaviour or interannual variability were derived. The obtained average D/L aspect ratios for T-matrix calculated cylindrical particles were 0.70 ± 0.20 and 1.90 ± 0.20 (similar to Wolff et al., 2001), and the best fitting Martian dust analogue corresponded to the basalt sample (in agreement with Dabrowska et al., 2015).

Mixed numerical and analytical method for investigating orbital angular momentum beam scattering in turbid water

A mixed numerical and analytical technique is presented to investigate orbital angular momentum (OAM) beam scattering in turbid water for underwater lidar applications. Electromagnetic simulations are used to generate single-scattering phase functions (SSPFs) that predict the angular scattering distribution for a single particle illuminated by either a Gaussian beam or an OAM beam. These SSPFs are used in array theory and radiative transfer calculations to predict the net volumetric scattering functions (VSFs) and transmittance for multiparticle scattering in a three-dimensional space for both Gaussian and OAM beams. Simulation results show that the VSFs (and therefore the transmittance) of Gaussian and OAM beams are nearly identical, with a slight dependence on OAM charge. Laboratory water tank transmission experiments are performed to verify the simulated predictions. The experimental results are in excellent agreement with the simulation predictions.

A novel phase function describing light scattering of layers containing colloidal nanospheres.

Light scattering from small particles exhibit unique angular scattering distributions, which are strongly dependent on the radius to wavelength ratio as well as the refractive index contrast between the particles and the surrounding medium. As the concentration of the particles increases, multiple scattering becomes important. This complicates the description of the angular scattering patterns, and in many cases one has to resort to empirical phase functions. We have measured the angle dependence of light scattering from a polymer layer containing sub-micron metallic and dielectric particles. The samples exhibited strongly forward and backward peaked scattering patterns, which were fitted to a number of empirical approximative phase functions. We found that a novel two-term Reynolds-McCormick (TTRM) phase function gave the best fit to the experimental data in all cases. The feasibility of the TTRM approach was further validated by good agreement with numerical simulations of Mie single scattering phase functions at various wavelengths and sizes, ranging from the Rayleigh scattering regime to the geometrical optics regime. Hence, the widely adaptable TTRM approach is able to describe angular scattering distributions of different kinds of nanospheres and nanocomposites, both in the single scattering and multiple scattering regimes.

Controlling optical memory effects in disordered media with coated metamaterials

Most applications of memory effects in disordered optical media, such as the tilt-tilt and shift-shift spatial correlations, have focused on imaging through and inside biological tissues. Here we put forward a metamaterial platform not only to enhance but also to tune memory effects in random media. Specifically, we investigate the shift-shift and tilt-tilt spatial correlations in metamaterials composed of coated spheres and cylinders by means of the radiative transfer equation. Based on the single-scattering phase function, we calculate the translation correlations in anisotropically scattering media with spherical or cylindrical geometries and find a simple relation between them. We show that the Fokker-Planck model can be used with the small-angle approximation to obtain the shift-tilt memory effect with ballistic light contribution. By considering a two-dimensional scattering system, composed of thick dielectric cylinders coated with subwavelength layers of thermally tunable magneto-optical semiconductors, we suggest the possibility of tailoring and controlling the shift-shift and tilt-tilt memory effects in light scattering. In particular, we show that the generalized memory effect can be enhanced by increasing the temperature of the system, and it can be decreased by applying an external magnetic field. Altogether our findings unveil the potential applications that metamaterial systems may have to control externally memory effects in disordered media.

A Fast All-sky Radiation Model for Solar applications with Narrowband Irradiances on Tilted surfaces (FARMS-NIT): Part I. The clear-sky model

The solar energy industry often uses individual steps to empirically compute plane-of-array (POA) irradiance from horizontal irradiance and decompose it to narrow-wavelength bands. Conventional radiative transfer models designed for meteorological applications requires significant computing efforts in practice; however, they provide a physics-based solution of radiance and therefore are capable of computing spectral POA irradiances in a single step. In this study, we integrate the advantages of the current models and develop an innovative radiative transfer model, the Fast All-sky Radiation Model for Solar applications with Narrowband Irradiances on Tilted surfaces (FARMS-NIT), to efficiently compute irradiances on inclined photovoltaics (PV) panels for 2002 narrow-wavelength bands from 0.28 to 4.0 µm. This study is reported in two parts. Part I presents the methodology and performance evaluation of the new model under clear-sky conditions. The Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS), which was designed to compute clear-sky irradiances, is employed to rapidly provide the optical properties of a given clear-sky atmosphere. The clear-sky radiances in the narrow-wavelength bands are computed by considering three paths of photon transmission and solving the radiative transfer equation with the single-scattering approximation. The Bi-directional Transmittance Distribution Function (BTDF) of aerosols is given by their single-scattering phase function with a correction using a two-stream approximation. The validation analysis confirms that FARMS-NIT has improved accuracy compared to TMYSPEC as evaluated by both surface observations and a state-of-the-art radiative transfer model. This model substantially improves computational efficiency compared to other radiative transfer models though it uses slightly more computing time than TMYSPEC. Part II of this study addresses the model in cloud-sky conditions and will be published as a companion paper.

Snow particles extracted from X-ray computed microtomography imagery and their single-scattering properties

Sizes and shapes of snow particles were determined from X-ray computed microtomography (micro-CT) images, and their single-scattering properties were calculated at visible and near-infrared wavelengths using a Geometrical Optics Method (GOM). We analyzed seven snow samples including fresh and aged artificial snow and natural snow obtained from field samples. Individual snow particles were numerically extracted, and the shape of each snow particle was defined by applying a rendering method. The size distribution and specific surface area distribution were estimated from the geometrical properties of the snow particles, and an effective particle radius was derived for each snow sample. The GOM calculations at wavelengths of 0.532 and 1.242 µm revealed that the realistic snow particles had similar scattering phase functions as those of previously modeled irregular shaped particles. Furthermore, distinct dendritic particles had a characteristic scattering phase function and asymmetry factor. The single-scattering properties of particles of effective radius reff were compared with the size-averaged single-scattering properties. We found that the particles of reff could be used as representative particles for calculating the average single-scattering properties of the snow. Furthermore, the single-scattering properties of the micro-CT particles were compared to those of particle shape models using our current snow retrieval algorithm. For the single-scattering phase function, the results of the micro-CT particles were consistent with those of a conceptual two-shape model. However, the particle size dependence differed for the single-scattering albedo and asymmetry factor.

Modeling and intercomparison of field and laboratory hyperspectral goniometer measurements with G-LiHT imagery of the Algodones Dunes

We compare field hyperspectral bidirectional reflectance distribution function (BRDF) measurements acquired by a hyperspectral goniometer system known as the goniometer of the Rochester Institute of Technology (GRIT) during an experiment in the Algodones Dunes system in March 2015 with NASA Goddard’s light detection and ranging, hyperspectral, and thermal imagery of the site acquired during the experiment. We augment our field spectral data collection with laboratory hyperspectral BRDF measurements of samples brought back from the Algodones Dunes site using GRIT and our second-generation goniometer GRIT-two (GRIT-T). In these laboratory experiments, we vary geophysical parameters such as sediment density and grain size distribution of the sediments that would typically impact observed BRDF with the goal of extending the range of applicability of our resulting BRDF spectral libraries. Geotechnical measurements on site confirm the variability of geophysical parameters such as density and grain size distributions within the dune system, and measurements with GRIT and GRIT-T demonstrate the impact on observed spectral variation. By augmenting field spectral libraries with laboratory BRDF, we show that a greater proportion of the dune system is more faithfully represented in the expanded spectral library. Beyond developing appropriate calibration data for airborne and satellite imagery of the Algodones Dunes, laboratory and field studies also support goals to develop reliable retrieval methods for geophysical quantities such as sediment density directly from spectral imagery. We consider approaches based on the Hapke model. Our approaches use the invariance of the observed functional forms of the single scattering phase function, which must be invariant to differences in the illumination geometry. Fill factor is retrieved and correlates with expected direct measurements of sediment density in a laboratory setting.

Aerosol properties in the upper clouds of Venus from glory observations by the Venus Monitoring Camera (Venus Express mission)

From the angular positions of the glory features observed on the upper cloud deck of Venus in three VMC channels (at 0.365, 0.513, and 0.965 µm), the dominating sizes of cloud particles and their refractive indices have been retrieved, and their spatial and temporal variations have been analyzed. For this, the phase profiles of brightness were compared to the single-scattering phase functions of particles of different sizes, since diffuse multiple scattering in the clouds does not move the angular positions of the glory, which is produced by the single scattering by cloud particles, but only makes them less pronounced. We presented the measured phase profiles in two ways: they were built for individual images and for individual small regions observed in series of successive images. The analysis of the data of both types has yielded consistent results. The presently retrieved radii of cloud particle average approximately 1.0–1.2 µm (though some values reach 1.4 µm) and demonstrate a variable pattern versus latitude and local solar time (LST). The decrease of particle sizes at high latitudes (down to 0.6 µm at 60°S) earlier found from the 0.965-µm and partly 0.365-µm data has been definitely confirmed in the analysis of the data of all three channels considered. To obtain the consistent estimates of particle sizes from the UV glory maximum and minimum positions, we have to vary the effective variance of the particle sizes, while it was fixed constant in our previous studies. The twofold increase of this parameter (from 0.07 to 0.14) diminishes the estimates of particle sizes by 10–15%, while the effect on the retrieved refractive index is negligible. The obtained estimates of the refractive index are more or less uniformly distributed over the covered latitude and LST ranges, and most of them are higher than those of concentrated sulfuric acid solution. This confirms our previous result obtained only at 0.965 µm, and now we may state that the cases of a relatively high real part of the refractive index are often observed for the 1-µm mode of cloud particles on Venus. Consequently, an additional component with a high value of the refractive index is required to be present in the cloud droplets. We suggest that this component is in small submicron particles; during the condensation process, they become incorporated into sulfuric acid droplets, which results in forming the complex UV absorbing particles with an increased refractive index. We suppose that this material can be ferric chloride that is one of the candidates for the so-called unknown UV absorber in the upper clouds of Venus.

Modified analytic expression for the single-scattering phase function

In electromagnetic radiative transfer calculation, the accuracy and the computation time are usually determined by the representation of single-scattering phase function. Accurate calculation is time consuming even for spherical particle, thus, an analytic representation is commonly adopted to approximate the exact phase function and then accelerate the calculation. Most widely used single-scattering phase functions are the Henyey-Greenstein phase function and modified Henyey-Greenstein phase function (Henyey-Greenstein*). Although the Henyey-Greenstein phase function and the Henyey-Greenstein* phase function can represent the forward-scattering peak of Mie-scattering phase function well, they fail to reproduce the backscattering behavior, limiting the accuracy of the calculation. In order to better fit exact calculations and simulate the backward-scattering peak, we develop a new analytic expression based on the fundamental theory of electromagnetic scattering and radiation transmission. This phase function is an algebraic expression with one single free parameter (asymmetry factor), and can be expanded into Legendre polynomials. The new phase function converges to the Rayleigh phase function when the asymmetry factor approximates to 0, and it can approach to the Henyey-Greenstein phase function as the asymmetry factor is about 1. We compare the Henyey-Greenstein phase function, the Henyey-Greenstein* phase function, and the new phase function for different asymmetry factors, and find that the new phase function provides a more realistic description for the unpolarized light scattering from small particles. Furthermore, the calculated value for the ratio of the scattering intensity at 90 degree to that in the backward direction is more reasonable. We also investigate the effectiveness by approximating the scattering from polydispersed particles through comparing the new phase function, the Henyey-Greenstein* phase function, and the Mie-scattering phase function for three types of Derimendjian's polydispersions. Results show that the new phase function fits the Mie-scattering phase function much better than the Henyey-Greenstein* phase function. For the new phase function, the root-mean-square error is small for 73.3% data. By contrast, only 26.7% data fit the Mie-scattering phase function well for the Henyey-Greenstein* phase function. Similarly, the effectiveness of new function is most significant when calculating the ratio of the scattering intensity at 90 degree to that in the backward direction. In summary, the new Henyey-Greenstein* phase function provides a more accurate calculation for the scattering intensity in the backward direction, and is conducive to electromagnetic radiative transfer calculation. Furthermore, because the proposed phase function has the same basic form as the Heny-Greenstein phase function, reformatting radiative transfer model in terms of the new phase function should require relatively little effort.

Effect of the single-scattering phase function on light transmission through disordered media with large inhomogeneities

We calculate the total transmission coefficient (transmittance) of a disordered medium with large (compared to the light wavelength) inhomogeneities. To model highly forward scattering in the medium we take advantage of the Gegenbauer kernel phase function. In a subdiffusion thickness range, the transmittance is shown to be sensitive to the specific form of the single-scattering phase function. The effect reveals itself at grazing angles of incidence and originates from small-angle multiple scattering of light. Our results are in a good agreement with numerical solutions to the radiative transfer equation.

Finite-size effect in light transmission through highly forward scattering media at grazing angles.

We present a theoretical study of light transmission through a disordered medium with large (compared to the light wavelength) inhomogeneities. Both numerical integration and analytic treatments of the radiative transfer equation are performed. An effect of the single-scattering phase function on the total transmittance is found in a subdiffusion thickness range. The effect reveals itself at grazing angles of incidence and originates from small-angle multiple scattering of light. A simple analytic formula for the total transmittance is derived. Our results are in good agreement with data of independent numerical calculations.