We propose a model to simulate the propagation of lightthrough an aerosol cloud. In this model, the aerosol cloudis divided into a succession of thin layers equivalent to“aerosol lenses”. These lenses affect only the scatteredlight; the unscattered light crosses them without otheralteration than attenuation. This approach is used inhigh-energy laser physics [1,2] to model the impact ofaerosols on a beam self-focusing mechanism. These mod-els do not propose to form an image of the light scatteredby the aerosols. We propose to extend these stratifiedmodels to produce the entire beam profile.Figure 1 (top) shows the configuration of the system to
reproduce. A narrow beam propagates through aerosols.Part of the initial beam will reach the screen unscatteredand part of it will be scattered by aerosols. The bottompart of Fig. 1 shows the new configuration of the system.The aerosols are replaced by multiple independent thinlayers. Each layer has a probability PS to scatter the lightand a probability PU to let it through.If a layer is thin enough, scattering order one domi-
nates all other scattering orders and it acts as a diverginglens with an “equivalent focal length” determined by theaerosol phase function. Taken independently, a layer willproduce a distinct scattered light image on the screen.This image will correspond to the aerosol phase function.Because individual layers are located at different dis-tances from the screen, they produce images having dif-ferent scale and intensity. This image represents the layercontribution to the scattering process. In the aerosol lensanalogy, it is the PSF. The Fourier transform of the PSF isthe MTF [3]. We therefore have the MTFs of a systemmade of n cascaded lenses. We can find the systemMTF knowing that the MTF of a systemmade of cascadedlenses is equal to the product of the MTFs of each lens[4,5]. The Fourier transform of the resulting MTFwill pro-vide the PSF of the whole system, i.e. the irradiance ofthe screen. This process is similar to a convolution.This model is based on two assumptions. The first as-
sumption is that each layer is dominated by order onescattering. The second assumption is that the light scat-tered outside the screen (including backscattered light)can be neglected.
Figure 2 shows the scattering mechanism in an indivi-dual layer. A light beam with power P0 arrives at thelayer. Inside the scattering layer, part of the light (PS)is scattered according to the aerosols phase function.We are interested only in the portion of the signal thatilluminates the screen within angle θ. The rest is ne-glected. Since the screen dimension is fixed, the valueof θ will change through propagation. For a layer closeto the screen we have θ ≈ π=2. The rest of the beam(PU) is unscattered. If we neglect the light scattered out-side θ we have P0 ¼ PU þ PS . Going further in simplifica-tion and posing P0 ¼ 1, PU is the probability to gothrough the layer unscattered and PS is the probabilityto be scattered. After n layers, we have the probabilitydistribution
P0 ¼ ðPU þ PSÞn: ð1ÞThe probability to be unscattered after n layer is Pn
U ;the probability to be scattered n times after n layer is gi-ven by Pn
S . The series expansion of Eq. (1) shows that thePS term to the ith power gives the probability to be scat-tered i times.
To find the final irradiance on the screen, we have togenerate the irradiance on the screen for each aerosollens. We will note this irradiance IiðrÞ, where i is thelayer and r the distance from the axis. The first step is
Fig. 1. (Color online) Conversion from a system propagating anarrow light beam through aerosols (top) to a stratified modelmade of a succession of “aerosol lenses” (bottom).
to generate ISiðrÞ, the scattered contribution to the irra-diance. Knowing the phase function Φðθ;φÞ of an aero-sol, the power P0 of the incoming beam, the distance dibetween the screen and the aerosol lens, the irradianceon the screen at radius r in W=m2 is given by
ISiðrÞ ¼P0ΦðθðrÞ;φÞ
d2icosðθðrÞÞ; ð2Þ
where θðrÞ ¼ arctanðr=diÞ. The term cosðθðrÞÞ is addedto take into account the incident angle of the light onthe screen. P0 is the same for all layers.Equation (2) considers only the scattered photons. The
photons going through the layer unscattered will form animage on the screen with profile IuðrÞ. This profile is thesame for all layers. Representing the narrow beam by a2D radial impulse function [6] and dividing by the area ofa resolution element we get an irradiance in W=m2,
IuðrÞ ¼P0
areaδðrÞπjrj : ð3Þ
Normalizing Eq. (2) and (3) at r ¼ 0 and using propor-tionality factors α and β yields the screen irradiancedistribution
IiðrÞ ¼ αiIsiðrÞIsið0Þ
þ βiIuðrÞIuð0Þ
: ð4Þ
The αi and βi factors are related to the power on thescreen and will be found later. Noting the Fourier trans-form of a single layer using HiðkÞ and by the additiontheorem [6], the normalized Fourier transform of an ade-quately balanced Eq. (4) is [7]
HiðkÞ ¼PsiHsiðkÞ þ PuiHuðkÞ
Pui þ Psi; ð5Þ
where Pui and Psi are the unscattered and scatteredpower on the screen for layer i and HsiðkÞ and HuðkÞare the normalized Fourier transform of the scatteredand unscattered contribution, respectively. The valueof Pui and Psi are given by
In Eq. (6), τi is the optical depth of the layer, P0 is theinitial power of the beam, and EsðθÞ is the encircled en-ergy within angle θ (Fig. 2) of the phase function given byEq. (7). Equation (5) is a properly balanced and normal-ized equation that can be used for convolution. Thereverse transform of Eq. (5) will give us a properlybalanced Eq. (4) with the correct values of αi and βi.
The convolution is the Fourier transform of the pro-duct of the HiðkÞ. It is noted
IðrÞ ¼ F
�Yni¼1
�PsiHSiðkÞ þ PuiHuðkÞ
Pui þ Psi
��: ð8Þ
Equation (8) gives the irradiance distribution on thescreen. Developing this model further, we can determinethe contribution of a specific scattering order. For thesimple case of an infinite screen located at a large dis-tance from the aerosols, HiðkÞ is the same for all layers.Going further in simplification and putting Pui þ Psi ¼ 1,we get
IðrÞ ¼ Fð½PsHsðkÞ þ PuHuðkÞ�nÞ: ð9ÞEquation (9) takes the form of Eq. (1), whose series
expansion was shown to provide individual scattering or-der contribution. Furthermore, it is known that convolu-tion of first-order processes will generate higher-orderprocesses [6]. To extract a specific order contribution,Eq. (8) has to be expanded. For a selection of scatteringorders we get
I0th ¼ F
�Yni¼1
�PuiHui
Psi þ Pui
��; ð10aÞ
I1st ¼ F
�Xnj¼1
�Hsj
Huj
Psj
Puj
Yni¼1
�PuiHui
Psi þ Pui
���; ð10bÞ
I2nd¼F
�Xn−1j¼1
Xnk¼jþ1
�HsjHskPsjPsk
HujHukPujPuk
Yni¼1
�PuiHui
PsiþPui
���; ð10cÞ
Inth ¼ F
�Yni¼1
�PuiHui
Psi þ Pui
��: ð10dÞ
We compared the results of the model to the Undiqueimaging Monte Carlo simulator [8]. This simulator isbased on the classic BHMIE algorithm [9]. Photons arelaunched individually in the simulator and reproduce
Fig. 2. (Color online) Image formation by a thin layer of aero-sols dominated by order one forward scattering.
all possible events (including backscattering contribu-tion). We made 12 simulations. In the simulated system,the source is a narrow 532 nm laser beam propagatingthrough 100m of a uniform aerosol made of waterdroplets 0:1 μm, 1 μm, 10 μm, or 100 μm in diameter.We simulated optical depth of 1, 2, and 3. Simulationshowed that an inhomogeneous distribution of layerswith numerous thin layers close to the screen yields bet-ter results than equally spaced layers. Consequently, theresults presented are calculated using 24 aerosol lenses.The first 15 lenses are located within 10m of the screenwith a geometric progression. Layer i is centered be-tween x ¼ ð1:585i − 1Þ=100 and x ¼ ð1:585i−1 − 1Þ=100and its optical depth is τi ¼ τ0ð1:585i − 1:585i−1Þ=100,where τ0 is the optical depth over a 1m range. The nineremaining lenses are spaced every 10m between 15mand 95m and have an optical depth τ ¼ 10τ0. In Fig. 3,we compare the irradiance predicted by the model (darklines) with the Undique results (red lines). In all cases,the screen is a 256 × 256 matrix. For the 10 μm and100 μm water droplets, one resolution element on thescreen is ð1 × 1Þ cm. For the 0:1 μm and 1 μm cases,one resolution element is ð8 × 8Þ cm. A larger screen cap-tures more energy and provides better results for thesesmaller aerosols. The results of the model and of the si-mulator match for all cases. The match is better for aero-sol 1 μm and larger because the amount of light scatteredoutside the screen is small and can be neglected.In Fig. 4, we show the simulations made with optical
depth τ ¼ 2 for different scattering orders. The MonteCarlo and the model agree within a few percents for par-ticles larger than 10 μm. With smaller particles, the dis-crepancy for high-scattering-order is large, as expected,
because the second assumption of the model is notrespected and multiple scattering is underestimated.
We have shown that the aerosol lenses model can re-produce the irradiance of a beam propagating throughaerosol clouds of various optical depths and size param-eters. The model can make an accurate prediction for aspecific scattering order when backscattered light can beneglected. We modeled simple systems, but using aerosollenses with various properties (optical depth, aerosolsize, location) we can model complex systems madeof several clouds or aerosols. The model has been ap-plied to reproduce the shower curtain effect for specificinstrumentation parameters. These results will make thesubject of a future publication shortly. The main advan-tage of this model is its execution speed. The model takesabout a second to generate one of the curves shown inFig. 3, while it takes several minutes to do so in theMonte Carlo.
References
1. E. P. Silaeva and V. P. Kandidov, Atmos. Oceanic Opt. 22,26 (2009).
2. V. O. Militsin, E. P. Kachan, and V. P. Kandidov, QuantumElectron. 36, 1032 (2006).
3. D. J. Schroeder, Astronomical Optics (Academic, 1987).4. R. E. Swing, Proc. SPIE 46, 104 (1974).5. J. B. DeVelis and G. B. Parrent Jr., J. Opt. Soc. Am. 57,
1486 (1967).6. R. N. Bracewell, The Fourier Transform and Its Applica-
tions (McGraw Hill, 1986).7. L. R. Bissonnette, Opt. Eng. 31, 1045 (1992).8. G. Tremblay, X. Cao, and G. Roy, Proc. SPIE 7828,
78280C (2010).9. C. F. Bohren and D. R. Huffman, Absorption and Scattering
of Light by Small Particles (Wiley-VCH, 1983).
Fig. 3. (Color online) Comparison between the stratified mod-el and the Undique Imaging Monte Carlo simulator. In the usualorder, the results for water droplets 100 μm, 10 μm, 1 μm, and0:1 μm in diameter. Red lines are the UNDIQUE results. Darklines are the aerosol lenses model results.
Fig. 4. (Color online) Scattered signal irradiance on the screenby scattering order for water droplets 100 μm, 10 μm, 1 μm, and0:1 μm in diameter at τ ¼ 2. Red lines are the UNDIQUE results.Dark lines are the aerosol lenses model results.
