#### Abstract

In this paper, we present a study of the influence of roughness on the bidirectional reflectivity and on the emissivity of surfaces using Maxwell’s electromagnetic theory. In this framework, we solve the Helmholtz equations by using the surface integral method. We first proceed to a general description of this method, allowing to solve the propagation equations of the electromagnetic field. We then express the diffused directional flux using the surface field and its normal derivative (source terms), in the case of an incident plane wave in “p” polarization or in “s” polarization. This finally allows us to arrive at the desired radiative properties. We have developed two numerical calculation codes whose use we limit to cases of surfaces presenting cavities in the shape of a symmetrical or asymmetrical “V”. Particular interest was given to the influence of the geometric parameters of these surfaces on the bidirectional reflection function and on the emissivity of these surfaces. Finally, we present some very conclusive results.

#### 1. Introduction

The theoretical or experimental determination of the radiative properties of rough surfaces is the subject of several research works. These parameters are involved in very varied fields of application ranging from the calculation of energy exchanges by thermal radiation to the design of selective rough surfaces, in addition to common applications in concentrated solar power, agriculture, medicine, radar waves, etc. A better understanding of the physical phenomena related to the interaction of electromagnetic waves with a surface is always necessary to master the problem. Different numerical simulation models to study these radiative properties have been developed. When the surface is made up of asperities with a characteristic dimension greater than the wavelength, it is said to be macrorough. In this case, among the study methods used, we cite the iterative method, the variational method, the Monte-Carlo statistical method, and the image method [1–4]. In the case of microrough surface, methods based on notions of physical optics are developed and applied [4]; a particular interest is then focused on the energy reflected in a coherent or incoherent way.

More recently, models based essentially on Maxwell’s equations and the integral surface method have made it possible to study the diffusion of electromagnetic waves by rough surfaces. In this context, we cite the work of Grefet [5], Greffet and Ladan [6], and Ladan and Buckius [7] as well as the diversified and highly enriching work of Buckius et al. [8–11] and Charles et al. [12]. The interest of these authors is particularly focused on the study of the bidirectional reflectivity or the emissivity of random or regular rough surfaces of a conductive or dielectric material. Most often, these surfaces are illuminated by a plane wave [13] or by an incident Gaussian beam [7]. The results are analyzed in the two cases of polarization “p” and “s,” that is, parallel and perpendicular. It should also be noted that experimental work is carried out in order to validate the various models developed [14–16].

In the present work, we propose to contribute to the study of the influence of a roughness in the shape of symmetrical or asymmetrical “V,” on the bidirectional reflectivity and on the emissivity of these surfaces using Maxwell’s electromagnetic theory. In this context, the Helmholtz equations are solved by the application of the surface integral method. We have limited the exploitation of these numerical calculation codes to the cases of surfaces presenting cavities in the shape of a symmetrical or asymmetrical “V.” The material constituting the studied surface being BaSO_{4}, with a complex refractive index corresponding to the wavelength .

Particular interest was given to the influence of the geometric parameters of these surfaces on the bidirectional reflection function and on the emissivity of these surfaces.

#### 2. Determination of Radiative Properties by using the Surface Integral Method

##### 2.1. Incident Field

We consider in our work a surface presenting longitudinal cavities contiguous in the form of symmetrical or asymmetrical “V.” In the section plane (xOz), the profile of this surface is represented by a periodic function of the variable *x*; this is characterized by its geometric period , its height *h,* and a parameter , between 0 and 1, indicating the position of the peak of the surface. These cavities are dug on the surface of a homogeneous, linear, and isotropic dielectric medium , with a complex refractive index *n*_{c}. We designate by the half-space defined by and assimilated to the void. We work with an incident plane wave at the angle in both cases of polarization. The first concerns the “p” polarization, for which the magnetic excitation field is perpendicular to the plane of incidence (xOz), and it is expressed in vacuum at a point *M* (*x*, 0, *z*) by:with:where H0 is the amplitude of the magnetic field and is the wave number.

The second case concerns the “s” polarization, for which the incident electric field is perpendicular to this plane. Case development will not be presented in this paper.

##### 2.2. Helmholtz Equations in “p” Polarization

We denote by the field evaluated at an observation point of coordinates *x* and *z* > , located in a vacuum. This field obeys the Helmholtz equation:and checks an outgoing wave condition at infinity.

The field in the medium evaluated at a point of coordinates *x* and obeys the equation:or . This field decreases exponentially when *z* tends to .

The surface passage conditions are as follows:where is the ordinate of the abscissa point *x*, which tends towards the surface while remaining in the middle . Whereas, is the ordinate of the abscissa point *x* which tends towards the surface while remaining in the middle . designates the partial derivative along the normal, to the surface at the point (*x*, *z* = ). The normal unit vector at the interface, oriented from towards , is defined as follows:with:the expression of the derivative following the normal is written as follows:

For an oriented normal of , the expression of the derivative is written as follows:

##### 2.3. Integral Surface Equations in “p” Polarization

###### 2.3.1. Integral Equations of the Field above the Interface

Having introduced Green’s functions [1,2], we now apply Helmholtz’s theorem to the differential equation (3). For this, we choose the volume limited by the closed surface and located above the interface . We can write for the field the following equations:where is the volume bounded by the closed surface and denotes the derivative along the outgoing normal. Subdivide the surface into two parts, one located just above the interface and the other closing on . The surface integral in (7) is then written as the sum of the integrals over the surfaces :

. By tending to a hemispherical surface of infinite radius located in , and taking into account the Sommerfeld condition [3], we get the following:

Using Cartesian coordinates, the expression for the integral (7) becomes:

Let us express the surface element dS. For this, we denote by *s* the curvilinear abscissa on the profile, and the elementary displacement on the latter is written as . The relations (7) and (8) expressing lead to:

Given (10), equation (14) can then be written in the following form:

We introduce at this level the source terms H and *L* defined by:

Physically, H (*x*) is the magnetic field at the coordinate point and L (*x*) represents, up to a constant, the tangential component to the surface of the electric field at this point. We finally arrive at the following equations:translating the integral expressions of surface of the field in the vacuum.

###### 2.3.2. Integral Equations of the Field below the Interface

A reasoning analogous to that used in the previous paragraph, applied to the volume limited by the surface located just below the interface and closing in , provides us with the two other equations which translate the field transmitted into the dielectric. To transform relation (4) into areal integral equations, we apply Helmholtz’s theorem, which allows us to write the following equations:

The surface is the meeting of two parts, one located just below the profile and the other closing on V. We tend towards a surface of infinite radius and located in the middle .

We will take into account that there is no incident field for and that the field transmitted in the dielectric verifies the Sommerfeld condition; so it comes:

It follows that the integral (20) can be written in the following form:or:

Using the continuity relations at the interface (5) and (6), this last equation is then written as follows:

By introducing the source terms given by relations (17) and (18), we end up with the following expressions:translating the surface integral equations of the field in the medium .

##### 2.4. Bidirectional Reflection Function

We show using equation (7), and the Fourier representation of [2] and taking , that the scattered magnetic field can be put in the following form:

By introducing the Poynting vector [4], and using relation (11), we express the diffused flux around (reflection angle) in the elementary angle :where is the width of the surface along the axis (oy) and is expressed by the following relation:

We deduce the two-way reflection function [5]:

A calculation analogous to that of the case of the “p” polarization allows us to write the bidirectional reflection function in “s” polarization:or:

Note that in this case of polarization, the source functions are defined by the following equation:

##### 2.5. Directional Monochromatic Emissivity

In the case of an opaque surface, knowledge of the hemispherical directional monochromatic reflectivity leads to that of the directional monochromatic emissivity from the relationship:

#### 3. Description of the Numerical Resolution Method

In order to calculate these source terms, we consider equations (26) and (27) by taking the observation point of coordinates (*x*, *z*) on the upper surface, that is, with an infinitely small real. The two integral equations (26) and (27) are then written, respectively, in the following forms:where:and where:

##### 3.1. Transformation of Integral Equations into Linear Systems

We propose to transform the two integral equations of the source terms into a linear system of equations where the unknowns are H and L. For this, we replace the infinite integrals by integrals limited to the interval . We divide the latter in N intervals of the same extent and whose centers have for abscissas given by:where:

It follows that equation (36) takes the following form:

By adopting the hypothesis of a very small variation of H (*x*) and L (*x*) on each of the intervals , we can approximate equation (41) by:

So taking we get:where:and:

We obtain:

Similarly, equation (37) takes the following form:with:and:

Equations (44) and (45) constitute a system with 2N equations and 2N unknowns with *n* ranging from 1 to N. The calculation of the coefficients of this system is presented in the appendix. It appears that:where denotes the Kronecker symbol, and with:and:where: et ,

The system of equations (46) and (47) therefore becomes:

It can be put in the following matrix form:

Or denote square blocks consisting of the respective matrix elements [2], and *H*, *L*, and , respectively, represent the N components .

A similar reasoning allows us to arrive at a linear system with 2N equations and 2N unknowns , components of the source terms in “s” polarization. The numerical resolution of the two linear systems makes it possible to arrive at the source terms in the two cases of polarization. These terms are necessary for the calculations of expressed by the relations (13) and (15). We replace these with the following expressions:

##### 3.2. Model Validation

###### 3.2.1. Validation in terms of Power

We carried out a validation of the developed model, based on a comparison between the ratio of the reflected power to the incident power, calculated using our model with that provided by the Fresnel formulas. As part of this validation, we consider the case of a plane wave and a plane surface for the two kinds of polarization. We note there that the average value along the surface of each of these polarizations is identical to that provided by the formulas of Fresnel (see Table 1).

###### 3.2.2. Validation in terms of Emissivity

We also find the curves provided by Dieenna and Buckius [16] in the case of both silicon and aluminum surfaces for various surface parameters (Figures 1 and 2).

The surfaces are constructed using a Fourier series representation from the surface. The number of terms in the series must be large enough for numerical convergence. However, the number of terms must be small enough to provide continuous derivatives numerically. Up to 75 terms have been included in the results presented. For the triangular surfaces shown, a 25-term Fourier series is used to generate the surface profiles. Typical surface lengths are divided into 2400 increments. These lengths require the memory limit of each machine. Surfaces that have a large require more increments than surfaces with small . All results presented conserve energy within 1 percent as evaluated by examining the conservation of energy for a dielectric surface (i.e., *K* = 0.0).

#### 4. Numerical Results and Interpretations Concerning a Rough Surface-Case of a “V” Surface Illuminated by a Plane Wave

We applied our model to the case of a surface presenting a “V” roughness and illuminated by a monochromatic plane wave of wavelength , under the angle of incidence , for the two polarizations “p” and “s”. This surface is made of barium sulphate (BaSO_{4}), with a complex refractive index of *n*_{C} = 1,628 + 0, 0003i, and it has cavities in the shape of a symmetrical or asymmetrical “V”. Recall that the profile of such a surface is characterized by its height *h*, its geometric period , and its peak position parameter . We are primarily interested in the behavior of the bidirectional reflection function with respect to the geometric parameters of the profile. Second, we examine the influence of the geometric period on the monochromatic directional emissivity of such a surface.

##### 4.1. Influence of Geometric Parameters on

For a height *h* equal to the wavelength of the incident radiation, and a value fixed at 0.5 of *f*_{0} (“V” symmetrical), we represent the variations of the bidirectional reflection function for values of the geometric period , respectively, equal to *λ*, 2 *λ*, 3 *λ*, and 4 *λ*, in each case of polarization. The representative curves show peaks of variations of different magnitudes, with, in each case, a predominant peak.

###### 4.1.1. Case of “p” Polarization

In the context of the “p” polarization, it appears from Figures 3(a)–3(d) that the representative curves of present a dominant peak of reflection corresponding to the angle of reflection . The amplitude of this peak goes from the value 0.12 to 0.55 when varies from *λ* to 3*λ*; while it takes the value 0.34, or 2.8 times that of the dominant peak relative to , for .

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The number of peaks increases when the geometric period increases, and it passes from two to seven peaks when passes from *λ* to 4*λ*. This result is consistent with that provided by the diffraction theory of gratings with the same geometric shape as our surface.

###### 4.1.2. Case of “s” Polarization

In the case of “s” polarization, Figures 4(a)–4(d) show that the geometric period also varies with the number of reflection peaks, their positions, and the amplitude of the dominant peak. The same variation in the number of peaks, as previously, is observed. The dominant reflection angle corresponds to the maximum amplitude peak, and it is −42° for values of equal to *λ*, 2*λ*, and 3 *λ*, while it is around 58° for . The value of is multiplied by 2.7 when going from *λ* to 2*λ* and by 5 when going to *λ* to 3*λ*; while it is equal to 0.75, 7 times that of the dominant peak relative to . The amplitude of the dominant peak is higher in the case of the “s” polarization than that corresponding to the “p” polarization.

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##### 4.2. Height Influence: h

We study the influence of the height *h* on the behavior of by considering the cases corresponding to the values of *h* equal to *λ*, 1.25*λ*, 1.5 *λ*, 1.75*λ,* and a geometric period equal to *λ*, in the two cases of polarization.

###### 4.2.1. Case of “p” Polarization

Figures 5(a)–5(d), in the case of the “p” polarization, essentially highlight the existence of two dominant reflection peaks, one located around the direction and the other located around of direction , symmetrical to the first with respect to the normal. The amplitude of these peaks increases with *h*. We report that the flow diffused in an incoherent way and grows when *h* increases. In addition, note that the amplitude of the dominant antispecular peak, that is, corresponding to the direction , takes precedence in all these cases over the specular peak defined by . Note that these observations remain practically valid when we consider a ratio less than unity, as shown in Figures 5(a)–5(d).

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###### 4.2.2. Case of “s” Polarization

In the case of the “s” polarization, the influence of the height *h* on is illustrated using Figures 6(a)–6(d). We note that there, as in the previous case, exist two reflection peaks, one following the specular direction and the other following the antispecular direction and there is an increase in incoherent diffused flux as *h* increases. However, for *h* equal to 1.25*λ* where 1.75*λ*, the amplitude of the specular reflection peak is greater than that of the antispecular peak.

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##### 4.3. Influence of the Asymmetry of the Cavity

To illustrate the influence of the asymmetry of the cavities on the bidirectional reflection function, we have varied the height *h* and the geometric period for values of *f*_{0} equal to 0.5, 0.7, and 0.9, in both cases of polarization. For the geometric period equal to *λ* and the height *h* equal to 1.25*λ*, we note that the positions of the peaks, located at 30° (specular) and at −30° (antispecular), remain unchanged when *f*_{0} varies. The amplitude of the antispecular peak increases remarkably when *f*_{0} goes from 0.5 (symmetrical “V”) to 0.9; whereas that of the specular peak starts increasing when *f*_{0} goes from 0.5 to 0.7, then decreases when *f*_{0} reaches 0.9. The flux reflected in an antispecular way becomes more and more important than that reflected in the specular direction when the asymmetry increases.

These results are illustrated in Figures 7(a)–7(c) and 8(a)–8(c). Note also that the same results are valid for a height *h* and a geometric period of the order of magnitude of the wavelength, in both cases of polarization. In Figures 9(a)–9(c) and 10(a)–10(c), we have represented the bidirectional reflection function for surfaces characterized by the parameters *h* equal to *λ* and equal to 4*λ*, in the same cases of asymmetry as previously described, in “p” and “s” polarization. We note that the reflected flux following a certain number of peaks in the case of symmetrical cavity concentrates along the normal when the asymmetry reaches the value 0.9.

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##### 4.4. Emissivity

###### 4.4.1. Influence of the Geometric Period

In Figures 11(a)–11(d), we represent the directional surface emissivity presenting cavities in the shape of a symmetrical “V” , of the same height *h* equal to *λ* and of respective geometric periods 4*λ*, 3*λ*, 2*λ*, and *λ*, in both cases of polarization. Note that for a smooth flat surface of the same material (BaSO_{4}), the emissivity is close to unity along directions up to 60° around the normal, and it then decreases for grazing directions. As we might expect, we see a slight increase in emissivity for directions up to grazing directions as the geometric period decreases. Indeed, this is consistent with the effect of the same roughness on the emissivity for a surface made of a material of the same nature [16].

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#### 5. Conclusion

This work allowed us to determine the radiative properties of rough surfaces from the electromagnetic theory by the surface integral method. The exploitation of computer codes, in the case of a triangular-shaped surface, led to the study of the influence of geometric parameters on the bidirectional reflectivity as well as on the emissivity of these surfaces. The results obtained are in an agreement with those provided by the theory of diffraction or by other research works in the treated case. Furthermore, we point out that the surface integral method becomes very expensive numerically as soon as the modulus of the complex refractive index of the medium increases.

#### Nomenclature

: | Incident flow |

: | Polarization reflection function “p” |

: | Reflection function in polarization “s” |

: | Volume located in |

V: | Volume located in |

: | Geometric period |

: | Dielectric medium |

: | Incident wave pulsation |

: | The void |

: | Surface profile |

: | Dielectric function |

λ: | Wave length |

: | Angle of incidence or reflection |

: | Dirac |

: | Normalization constant |

: | Bidirectional reflection function in “s” polarization |

: | Magnetic permeability of the medium |

: | Vacuum permeability |

: | Vacuum permittivity |

: | Electric field |

: | Magnetic excitation field |

: | Peak position of the “V” surface profile |

G: | Green’s function |

: | Zero-order Hankel function of the first kind |

: | Conjugate of the magnetic field |

h: | Height of the “V” surface profile |

: | Wave vector |

n_{C}: | Complex index |

p; TM: | Magnetic transverse polarization |

s, TE: | Electrical transverse polarization |

: | The stream diffused through the plane z = constant located above the profile |

: | Closed surface that limits the volume |

: | Firm surface which limits the volume V |

: | Mean value of the Poynting vector |

: | Bidirectional reflection function in polarization “p”. |

#### Data Availability

Data are available on request.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.