**2. Optical properties**

#### **2.1 Description of the optical properties**

For the description of the physical phenomenon of propagation of a pulse of light through a turbid media (energy transfer), the balance of incoming, outgoing, absorbed, and emitted photons is used. This is a wide, established, and well-known field, with a dense literature corpus [17–23].

Adverse weather conditions can be thought of as turbid media, as particles of different types and shapes (water, smoke, dust … ) are found suspended within the main media, which is air [24]. By definition, a turbid medium is characterized by having localized non-uniformities randomly distributed within it. These optical nonuniformities are usually inclusions of one substance within another with a different index of refraction *n*. These inclusions cause the medium to be optically inhomogeneous and cause it to behave as a scattering media.

In the air, which is a non-absorbing media, the inclusions, widely represented as "particles," are in charge of the actual absorption of part of the propagating energy and the actual change of direction of light. As a result, the dominant effects in the medium are absorption and scattering. According to these two effects, the medium is represented by several key optical parameters: the absorption coefficient *μa*, the scattering coefficient *μs*, the scattering phase function *p*ð Þ *θ*, *ϕ* , and the asymmetry factor *g*; which respectively describe: the absorbing and scattering power, the probability of scattering in a particular direction ð Þ *θ*, *ϕ* of the media, and the degree of scattering in the forward direction.

An absorbing medium is composed of particles (or other structures) that can absorb light and transform it into its internal energy as the beam is propagated along the medium, which results in a gradual reduction of the light intensity. To characterize this phenomenon, one uses a parameter that is called the absorption coefficient *μ<sup>a</sup>* (in units of *length*�<sup>1</sup> , usually in *cm*�<sup>1</sup>Þ, which quantifies the absorbing effect of the medium.

The deviation of light from its straight trajectory due to localized non-uniformities in the medium is known as scattering. The particles become scattering centers, which, when exposed to light, modify the electromagnetic field and re-emit it in a different direction. Analogously to the absorption case, a parameter called the scattering coefficient *μ<sup>s</sup>* (units of *length*�<sup>1</sup> , typically *cm*�1<sup>Þ</sup> is defined, which quantifies the scattering effect.

When a collimated beam of light passes through a volume of the medium, it will lose intensity due to both processes: absorption and scattering. In general, both processes cannot be distinguished. This effect is characterized by what is known as the extinction or attenuation coefficient *<sup>μ</sup><sup>t</sup>* <sup>¼</sup> *<sup>μ</sup><sup>a</sup>* <sup>þ</sup> *<sup>μ</sup><sup>s</sup>* (*cm*�1Þ. The extinction coefficient measures the total loss of a narrow-beam intensity, i.e. the loss due to absorption and the loss corresponding to the part of photons that have not been scattered in the forward direction.

Along with the scattering coefficient, a scattering event needs other parameters to be completely defined. In particular, photons may not be isotropically scattered and may need to have this dependence characterized. To completely define the deflection of the trajectory in space after a scattering event, two angles are used. The deflection scattering angle *θ* (which ranges from 0 to *π*) defines the deflection of the trajectory in the scattering plane—the plane formed by the direction of the incident light and the direction of the outgoing scattered light, i.e., the cone angle; and the azimuthal angle ϕ(which is defined from 0 to 2*π*) defines the change in the plane perpendicular to the scattering plane. These angles are shown in **Figure 2**, where the geometry of a scattering event is schematically depicted.

The directionality of the scattering effect is quantified using a phase function *p*ð Þ θ, ϕ . The phase function corresponds to the angular distribution of the light scattered by a scattering center at a given wavelength. It can be thought of as a

**Figure 2.** *Schematics of the scattering geometry.*

probability density function, showing the chances of a photon being scattered in a particular direction.

When the suspended particles in media have no preferred scattering orientation (spherical symmetry), it is known as an isotropic medium. Then, light is scattered equally in all directions. However, it is more frequent that natural materials scatter light preferentially in the backward or forward direction. For those non-isotropic cases, it is interesting to know the amount of energy retained in the forward/backward direction after a single scattering event. If a photon is scattered so that its trajectory is deflected by a certain deflection angle *θ*, then the component of the new trajectory, which is aligned in the former forward direction, is presented as cosð Þθ . The mean value of this cosine is known as the anisotropy factor *g*, and it is defined as:

$$\mathbf{g} \equiv \langle \cos \theta \rangle = \int\_0^\pi p(\Theta) \cos \Theta \cdot 2\pi \sin \Theta \, d\Theta \tag{1}$$

Its value varies in the range from �1 (total backward scattering) to 1 (total forward scattering), being *g* ¼ 0 the value corresponding to isotropic scattering.

In addition, scattering may be elastic and inelastic. Without entering into many details, elastic scattering is associated with an interaction with no energy losses (and, thus, no wavelength change), while inelastic scattering corresponds to a process with energy transfer and thus to a wavelength shift. Generally speaking, elastic scattering is the predominant effect when propagating through turbid media, as approximately only one in each 10<sup>7</sup> photons is inelastically scattered. Elastic interactions between photons and scattering particles are mainly described using two physical models: the Rayleigh and the Mie theories. The description of the process using one or another model is linked to the particle size and the wavelength of the incident light. However, Mie Theory is a general model developed using Maxwell equations and gives exact solutions in all cases, which means that it could be valid for any particle size.

Generally speaking, a turbid medium is described as a system of discrete spherical particles suspended within a base medium, which is exact for the case of fog (water suspended in air) and an approximation for the rest of the cases mentioned for the atmosphere (smoke … ). Such spherical particles enable Mie Theory to be used for its characterization. This theory gives quantitative results of the interaction of an electromagnetic plane wave with a single homogeneous sphere, being likely the most important exactly soluble problem in the theory of absorption and scattering by small particles. Mie Theory allows the calculation of the absorption and scattering coefficients and the phase function of a spherical particle of radius *a* as a function of the incident radiation wavelength *λ*, the size parameter *x* ¼ 2*πa=λ*, and the complex refractive indexes of the particles and the host material. The derivation of the complete theory may be long and tedious, and detailed information can be found in [19, 25].

#### **2.2 Mie theory**

The interaction of light with a spherical particle can be described and quantified using Mie Theory. Some conditions, however, have to be fulfilled to apply the theory. It has to be supposed that the media is homogeneous and that the particles that are embedded within it are spherical, homogeneous, and act independently—so they are distant enough from each other to consider only far-field scattering effects. Their radius and refractive index need to be known. As with most problems in theoretical

optics, the scattering of light by a homogeneous sphere is treated as a formal problem of Maxwell's theory with the appropriate boundary conditions [19, 25].

Suppose that one or more particles are placed in a beam of electromagnetic (EM) radiation. The rate at which EM energy is received by a detector downstream from the particles is denoted by *U*. If the particles are removed, the power received by the detector is *U*0, where *U*<sup>0</sup> > *U* . We say that the presence of the particles has resulted in the extinction of the incident beam. If the medium in which the particles are embedded is non-absorbing (such as air), the difference *U*<sup>0</sup> � *U* accounts for absorption and scattering by the embedded particles (water droplets). Although the specific details of extinction depend on many parameters, certain general features are shared in common by all particles.

Now, consider extinction by a single arbitrary particle embedded in a nonabsorbing medium and illuminated by a plane wave. If an imaginary sphere of radius r is constructed around the particle, the net rate at which EM energy crosses the surface *A* of the sphere is *WA*. If *WA* <0, energy is absorbed within the sphere (being *WA* the rate at which energy is absorbed by the particle).

*WA* maybe conveniently written as the sum of:

$$W\_A = -W\_S + W\_{ext} \tag{2}$$

*WS* is the rate at which energy is scattered across the surface *A*, and, therefore, *Wext* is just the sum of the energy absorption rate and the energy scattering rate: *Wext* ¼ *WS* þ *WA*.

Now it is possible to define *Cx* as the ratio of *Wx* (being *x*: *A*, *S* or *ext*) to *Ii* (incident irradiance):

$$\mathbf{C}\_{\mathbf{x}} = \frac{\mathbf{W}\_{\mathbf{x}}}{I\_{\mathbf{i}}} \tag{3}$$

The *Cx* quantities are called cross sections of the particle, and they have area dimensions. Let the total energy scattered in all directions be equal to the energy of the incident wave falling on the area *CS*; likewise, the energy absorbed inside the particle may be defined as the energy incident in the area *CA*, and the energy removed from the original beam may be equal to the energy incident in the area *Cext*, which gives an idea of the amount of energy removed from the incident field due to scattering and/or absorption generated by the particle. The law of conservation of energy then requires that:

$$\mathbf{C}\_{\text{ext}} = \mathbf{C}\_{\text{A}} + \mathbf{C}\_{\text{S}} \tag{4}$$

When solving Maxwell's equations for the defined problem, the scattered EM field is written as an infinite series in the vector spherical harmonics *Mn* and *Nn*, which are the EM normal modes of the spherical particle. Thus, the scattered field is expressed as a superposition of these normal modes, each weighted by the appropriate coefficient *an* or *bn*, known as scattering coefficients.

It is found that:

$$\mathbf{C}\_{S} = \frac{W\_{s}}{I\_{i}} = \frac{2\pi}{k^{2}} \sum\_{n=1}^{\infty} (2n+1) \left( |a\_{n}|^{2} + |b\_{n}|^{2} \right) \tag{5}$$

$$C\_{\rm ext} = \frac{W\_{\rm ext}}{I\_i} = \frac{2\pi}{k^2} \sum\_{n=1}^{\infty} (2n+1) \operatorname{Re} \left\{ a\_n + b\_n \right\} \tag{6}$$

*Modeling the Use of LiDAR through Adverse Weather DOI: http://dx.doi.org/10.5772/intechopen.109079*

$$\mathbf{C}\_{\text{A}} = \mathbf{C}\_{\text{ext}} - \mathbf{C}\_{\text{S}} \tag{7}$$

Assuming that the series expansion of the scattered field is uniformly convergent, it is proved that the series can be terminated after:

$$m = \varkappa + 4\varkappa^{1/3} + 2\tag{8}$$

where *x* ¼ *ka* is the size parameter, with *k* being the wavenumber and *a* the radius of the sphere.

We also need to obtain explicit expressions for the scattering coefficients:

$$a\_n = \frac{\mathcal{S}\_n'(\boldsymbol{\mathcal{y}})\mathcal{S}\_n(\boldsymbol{\mathfrak{x}}) - m\mathcal{S}\_n(\boldsymbol{\mathfrak{y}})\mathcal{S}\_n'(\boldsymbol{\mathfrak{x}})}{\mathcal{S}\_n'(\boldsymbol{\mathfrak{y}})\mathcal{L}\_n(\boldsymbol{\mathfrak{x}}) - m\mathcal{S}\_n(\boldsymbol{\mathfrak{y}})\mathcal{L}\_n'(\boldsymbol{\mathfrak{x}})} \tag{9}$$

$$b\_n = \frac{mS\_n'(\boldsymbol{\nu})S\_n(\boldsymbol{\kappa}) - S\_n(\boldsymbol{\nu})S\_n'(\boldsymbol{\kappa})}{mS\_n'(\boldsymbol{\nu})\zeta\_n(\boldsymbol{\kappa}) - S\_n(\boldsymbol{\nu})\zeta\_n'(\boldsymbol{\kappa})} \tag{10}$$

where:

$$\mathcal{S}\_n(\mathbf{z}) = \sqrt{\frac{\pi z}{2}} I\_{n+0.5}(\mathbf{z}) \tag{11}$$

$$
\zeta\_n(\mathbf{z}) = \sqrt{\frac{\pi \mathbf{z}}{2}} H\_{n+0.5}^{(2)}(\mathbf{z}) \tag{12}
$$

With *Jn*þ0*:*5ð Þ*z* being the half-integral-order spherical Bessel function of first kind and *H*ð Þ<sup>2</sup> *<sup>n</sup>*þ0*:*5ð Þ*z* the half-integral-order Hänkel function of the second kind. The variables *x* and *y*, in this case, correspond to *x* ¼ *ka* and *y* ¼ *mka*, and *m* is the relative refractive index between the sphere and the medium in which it is embedded; and finally, *S*<sup>0</sup> *<sup>n</sup>*ð Þ*z* and *ζ*<sup>0</sup> *<sup>n</sup>*ð Þ*z* denote the derivatives of the corresponding functions.

Once the cross section of a single interaction has been computed, one needs to characterize the media. *μ<sup>a</sup>* can be understood as the sum of contributions of the absorption cross sections of the absorbers per unit volume, i.e., the product of the absorption cross section *CA*(*cm*2) by the density of absorbers *ρ<sup>a</sup>* (#*=cm*�3):

$$
\mu\_a = \mathbf{C}\_A \rho\_a \tag{13}
$$

Analogously, knowing the number of scattering particles per unit volume *ρ<sup>s</sup>* (#*=cm*�3) and their *CS*, it is possible to compute the scattering coefficient *μ<sup>s</sup>* of the propagating medium as:

$$
\mu\_{\mathfrak{s}} = \mathsf{C}\_{\mathbb{S}} \rho\_{\mathfrak{s}} \tag{14}
$$

The phase function and the asymmetry factor can also be computed using Mie Theory:

$$p(\theta) = \frac{2\pi}{k^2 C\_\mathcal{S}} \left( |\mathcal{S}\_1|^2 + |\mathcal{S}\_2|^2 \right) \tag{15}$$

$$\log \frac{2\pi}{k^2 \mathcal{C}\_{\mathcal{S}}} \left( \sum\_{n=1}^{\infty} \frac{2n+1}{n(n+1)} \operatorname{Re} \left\{ a\_n b\_n^\* \right\} + \sum\_{n=1}^{\infty} \frac{n(n+2)}{n+1} \operatorname{Re} \left\{ a\_n a\_{n+1}^\* + b\_n b\_{n+1}^\* \right\} \right) \tag{16}$$

being *S*<sup>1</sup> and *S*2:

$$S\_1 = \sum\_{n=1}^{\infty} \frac{2n+1}{n(n+1)} (a\_n \pi\_n \cos \theta + b\_n \pi\_n \cos \theta) \tag{17}$$

$$S\_2 = \sum\_{n=1}^{\infty} \frac{2n+1}{n(n+1)} (b\_n \pi\_n \cos \theta + a\_n \pi\_n \cos \theta) \tag{18}$$

From which *an* and *bn* are the coefficients computed in Eqs. (9) and (10), and *π<sup>n</sup>* and *τ<sup>n</sup>* two angle-dependent functions known as Mie angular functions. These angular functions are generated with the associated Legendre polynomials, and they can be calculated from the recurrence relations:

$$
\pi\_n = \frac{(2n-1)\mu}{n-1}\pi\_{n-1} - \frac{n}{n-1}\pi\_{n-2} \tag{19}
$$

$$
\pi\_n = n\mu\pi\_n - (n+1)\pi\_{n-1} \tag{20}
$$

where μ ¼ cosθ, and the first terms of π<sup>n</sup> are π<sup>0</sup> ¼ 0 and π<sup>1</sup> ¼ 1.
