REGULAR PAPER

Scattering of electromagnetic waves from random rough metallicsurfaces with one-dimensional structure calculated by the surfaceimpedance boundary condition method (SIBC)

Mehdi Khemiri1 • Imed Sassi1 • Mohamed Oumezzine1

Received: 12 February 2015 / Accepted: 22 June 2015

� The Optical Society of Japan 2015

Abstract A comparison between exact solutions and

surface impedance boundary condition method, for the

phenomenon of light scattering from one-dimensional

random rough metallic surfaces in both cases of polariza-

tion (s- and p-) is presented. The reflective properties of

random rough metallic surfaces at a large angle of inci-

dence have been reported on due to their potential appli-

cations in some of the radiative heat transfer research areas.

The influence of surface roughness, shadowing effects, the

nature of polarization, and the nature of the material on the

behavior of reflectivity and emissivity has been quantified.

1 Introduction

The evaluation of the electromagnetic waves scattering on

randomly rough surfaces has been used in many applica-

tions such as remote sensing, telecommunications, optics,

medical imagery, photorealistic image generation in com-

puter graphics, etc. The calculation of the distribution of

the intensity of light scattered from randomly rough sur-

faces has been reported using several exact and approxi-

mate models. Exact models based on the extinction

theorem [1–3] have been used to resolve Maxwell’s

equations at the surface materials for a wide range of

surface roughness parameters. These models have been

successfully applied to dielectric, metallic or perfectly

conducting surfaces [2–12].

Among these methods, we have, rigorous coupled wave

analysis (RCWA) which has been used to calculate spectral

properties of metallic grating [13–16]. Recently, Qiu et al.

have reported on the reflective properties of randomly

rough metallic surfaces at large incidence angle [17]. Sai

et al. in [18] used the finite-difference time domain method

(FDTD) to investigate the spectral properties of one-di-

mensional tungsten grating.

Exact methods present many disadvantages. They are

very expensive. However, the interpretation of the physical

mechanisms involved in the scattering process is difficult

because different effects, e.g., shadowing or multiple

scattering, are included implicitly in the calculation, and

there is no way of separating them out [19]. Moreover,

these methods are difficult to implement for surfaces in two

dimensions because of requirements on computer memory

and calculation time.

Approximate models, based on simplified assumptions,

have been widely used to treat the problem of the scattering

phenomena for different reasons. These models, explicitly,

enable to take into account certain surface characteristics

(root mean square of height, correlation length, and curva-

ture) and allow to break down the local and non-local inter-

actions involved in the process of electromagnetic waves

scattering. Moreover, these approximate models are very

easy to implement and require short time calculations.

Among these approximate models, we have the small per-

turbation method with first order (SPM1) which requires, in

the case of deterministic surfaces, the knowledge of Fourier

Transform of the surface height. Kirchhoff’s approximation

(KA), Geometric Optics Approximation (GOA), Small Slope

Approximation (SSA), are computationally less expensive

than exact methods. However, many authors have used these

methods, and studied the domains of validity of these meth-

ods by comparing them with exact methods [4, 20–28].

& Mehdi Khemiri

khemirifsm@gmail.com

1 Laboratoire physic-chimie des Materiaux-Faculte des

sciences de Monastir, Universite de Monastir, 5019 Monastir,

Tunisia

123

Opt Rev

DOI 10.1007/s10043-015-0116-3

Electromagnetic waves scattering by random rough

metallic surfaces has been studied by several authors. J. A.

Sanchez and M. Nieto-Vesperinas have reported on the

resonance effects in multiple light scattering from statisti-

cally rough metallic surfaces using a Monte Carlo simu-

lation method. They studied intensity, as well as the effects

associated with the excitation of surface electromagnetic

waves, and, in particular with surface polaritons (SP) [29].

M. Saillard and D. Maystre in [3] reported on the phe-

nomenon of scattering from metallic and dielectric rough

surfaces. Maradudin et al. in [6] used Green’s second

integral theorem to investigate the scattered electromag-

netic field from random metallic grating. For the finite

conductivity metal, it is shown that exact methods are

computationally very intensive and not suitable to metallic

surfaces because of the complex structure of metallic sur-

face. Therefore, an alternative approach called surface

impedance boundary condition (SIBC) was proposed [30–

33]. This method has been used for over 74 years as a

means of analyzing the problems of reflection from con-

ducting surfaces. It is used to calculate the efficiency of a

metallic diffraction grating [34, 35] and grating of small

curvature [36].

In this paper, we report on the numerical study of

scattering from random rough metallic surfaces using SIBC

method. We are interested in studying the variation of the

bidirectional reflectivity (BDR) of two different surface

materials (gold and silver). The effects of roughness, the

nature of polarization, incidence angle, and surface mate-

rials on the behavior of the radiation properties are pre-

sented. The regions of validity of SIBC method are

discussed in comparison with results from literature.

2 Surface geometry

We consider a one-dimensional rough surface. The surface

profile function in the x–z plane Pr[z = h(x)] separates a

vacuum region V1[z[ h(x)] from a metal medium

V2[z\ h(x)]. The surface is only varying in the x direction,

so that scattering is considered in the plane of incidence,

and it has a Gaussian random process which is completely

characterized by the correlation function defined by:

h x1ð Þh x2ð Þh i ¼ r2C x1 � x2ð Þ ¼ r2 exp � x1 � x2ð Þ2

s2

!

ð1Þ

where r is the root mean square (RMS) height and s is thecorrelation length. The Fourier transform of r2C(x) is the

power spectral density W(kx). A surface of size L is to be

generated from the power spectral density. We make

h(x) periodic outside, i.e.,

h(x) = h(x ? L). A Fourier series is used to represent

h(x) is given by

h xð Þ ¼ 1

L

X1n¼�1

bn kxnð Þ exp� ikxnxð Þ ð2Þ

where bn(kxn) = 2p(LW(kxn)) is a Gaussian random vari-

able, and kxn = 2pn/LTo model rough surfaces, we use a Gaussian correlation

function given by:

C xð Þ ¼ r2 exp � x2

s2

� �ð3Þ

The corresponding power spectral density is:

W kxð Þ ¼ r2s2ffiffiffip

p exp � k2xs2

4

� �ð4Þ

Random rough surface scattering were initially modeled by

Gaussian correlation functions in Monte Carlo simulations

[37–39].

Figure 1 shows the scattered electromagnetic waves

from 1-D Gaussian rough surface with Gaussian correlation

function. e is the permittivity of air, and e(x) is the per-

mittivity of material. ki, ks, hi, and hs, designate, respec-tively, an incident field, a scattered field, an angle of

incidence, and scattering angle.

3 Integral formalism for rough surfaces

Let U1 exp (-jxt) represent the y component of the electric

field in the s-polarization case or the magnetic field in the

p-polarization case. The indices inc, 1, and 2 designate,

respectively, an incident field, a total field component

above the surface, and that below the profile of surface.

The incident field is

Uincðx; zÞ ¼ expðj ða0x� b0zÞÞ; ð5Þ

where a0 = k0 sin h0, b0 = (k02 - a0

2)1/2and k0 = 2p/k.The total field y component is a solution of Helmholtz

wave equations in air (or vacuum) region,

Fig. 1 Rough surface scattering geometry

Opt Rev

123

r2U 1ð Þ x; zð Þ þ k20Uð1Þ x; zð Þ ¼ 0; z[ h xð Þ; ð6Þ

and in the metal region,

r2U 2ð Þ x; zð Þ þ k20e xð ÞUð2Þ x; zð Þ ¼ 0; z\h xð Þ; ð7Þ

and it satisfies the boundary conditions

Uð1Þðx; zÞÞ��z¼hþðxÞ¼ Uð2Þðx; zÞÞ

��z¼h�ðxÞ; ð8Þ

oUð1Þðx; zÞon

����z¼hþðxÞ

¼ 1

C

oUð2Þðx; zÞÞon

����z¼h�ðxÞ

; ð9Þ

as well as radiation condition at the infinite z (outgoing

waves at z ? ±?). q/qn indicates the normal derivative,

and C is a constant equal to 1 (s-polarization) or e (p-

polarization). The superscript on h? indicates that the field

and its normal derivative are evaluated in the limit

approached from above, and analogous comments hold for

h-.

By the use of Green’s second theorem and boundary

conditions we can derive the integral equations for the

scattered electromagnetic field. This field is expressed in

terms of the unknown field and its normal derivative

(source functions); these functions are defined in vacuum

on the boundary (Pr) as follows:

-for p-polarization case

HðxÞ ¼ Hð1Þðx; zÞ��z¼hþðxÞ; ð10Þ

LðxÞ ¼ �h0ðxÞ o

oxþ o

oz

� �Hð1Þðx; zÞ

�����z¼hþðxÞ

; ð11Þ

-for s-polarization case

EðxÞ ¼ Eð1Þðx; zÞ��z¼hþðxÞ; ð12Þ

FðxÞ ¼ ð�h0ðxÞ o

oxþ o

ozÞEð1Þðx; zÞ

����z¼hþðxÞ

; ð13Þ

where h0(x) is the derivative of h(x).

From integral equations giving a field at any point M(x,

z), one can obtain a pair of coupled integral equations for

the source functions for both polarizations. The resolution

of these equations, as described in references [6, 43] con-

sists in converting the infinite systems of integral equations

into two finite systems of linear equations as shown, in

turn, in Refs. [5] and [40].

4 Description of the SIBC method

Somemetals used in optics, such asAl, Ag, andAu has a high

conductivity. Although when the values of the refraction

index of the metals increase, that yields the resolution of

coupled integral equations for the source functions become

very hard because of the appearance of the term |n*| on the

argument of the Hunkel function. This problem can also be

solved if the relation between the tangential components of

the fields E// and H// along the surface is known. In this

second approach, Maxwell’s boundary conditions are

replaced by a boundary condition of the form:

E== ¼ Z n~ x H==

� �; ReðZÞ[ 0; ð14Þ

where n~ is a normal unit vector pointing towards the

exterior of the scatter and Z is a second rank tensor called

the surface impedance tensor. If Z is known the fields

outside the scattered can be found without solving Max-

well’s equation inside it.

Using the boundary conditions (Eqs. 8–9) and the inte-

gral equations for Refs. [5, 38, 39], the surface impedance

is:

Zs�pol ¼ k

j

EðxÞdEðxÞdn

: ð15Þ

By using the definition of F given by the Eq. (13), we get

Zs�pol ¼ k

j

EðxÞc FðxÞ ; c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h0 2ðxÞ

p: ð16Þ

In the p-polarization, we get:

Zp�pol ¼ j

k

dHðx;hðxÞÞdn

Hðx; hðxÞÞ : ð17Þ

By using the definition of L given by the Eq. (11), we get

Zp�pol ¼ j

k

c LðxÞHðxÞ : ð18Þ

We define two operators (G and N) associated to the

function of Green and its normal derivative, respectively.

The application of such operators to the function / is

represented as:

G /ð Þ ¼Zþ1

�1

G x; x0ð Þ/ x0ð Þdx0 and

Nð/Þ ¼Zþ1

�1

dGðx; x0Þdn

/ðx0Þ dx0:

ð19Þ

Using the delta family as |n*| ? ? the kernel G(x, x0)tends to a delta function G(/) can be approximated in the

following way:

Gð/Þ � /ðxÞZþ1

�1

Gðx; x0Þ dx0: ð20Þ

In the approximation, it is found that the operators G and N

tend towards multiplicative operators [34–36, 44].

Opt Rev

123

Gð/Þ ¼ /ðxÞ2 j k n � cðxÞ and Nð/Þ � j

4kn�h00ðxÞ /ðxÞ

c3ðxÞ :

ð21Þ

When the above relations are introduced into the integral

equations [44], we get:

1þ jh00

2k n � q3

� �HðxÞ ¼ jn�

kq3LðxÞ; p-polarization ð22Þ

1þ jh00

2k n � q3

� �EðxÞ ¼ j

kn � q3 FðxÞ s-polarization

ð23Þ

q is the radius of curvature of the random rough surface

z = h(x), qðxÞ ¼ Z01þh02ðxÞð Þ

32

h00ðxÞ : By using the Eqs. (16) and

(18) the surface impedance become:

Zp�polðxÞ ¼ Z0 1þ j

2k n � qðxÞ

� ; ð24Þ

Zs�polðxÞ ¼ Z0 1þ j

2k n � qðxÞ

� �1

; ð25Þ

where Z0 = 1/n* is the constant value obtained for flat

boundary [45].

Using the Poynting theorem relation, [46, 47] the bidi-

rectional reflection function and the bidirectional trans-

mission function are stated in Refs [5, 38, 39]. The

hemispherical radiative properties of reflection and emis-

sion are obtained by integrating the bidirectional functions

[5, 38, 39, 42].

It is shown that SIBC method is more suitable than exact

methods. This method greatly reduces the storage and

computational requirements for the problem of scattering

by random rough metallic surfaces.

5 Results and discussion

The results presented in this section are for scattering from a

surface of length L = 25.6k, where k is the wavelength. Thesurface is discretized into at 1024 points in x, depending on

the roughness of the surface. The roughness parameters are

r and s. The incident beam is p- or s-polarized at a given

angle of incidence hi. The result in both figures is an averageof the results for 300 surface realizations.

The influence of the angle of incidence, the nature of

polarization, and surface materials is interesting. As a

check on the validity of the SIBC method, the results were

compared with numerical [6] results published previously.

In Fig. 2 we compare the results of the present method

and numerical results taken from ref [6]. In this figure, we

are interested in the variation of the bidirectional

reflectivity (BDR) as a function of the scattering angle hsfor p-polarized light incident on a random rough metallic

surface characterized by the statistical parameters: RMS

height r = 1.4142 lm and a correlation length s = 2 lm.

The wavelength of the incident light was k = 0.6127 lm.

The surface material is the silver with dielectric constant at

this wavelength e(x) = -17.2 ? i0.498. What is reported

here is the behavior of the enhanced backscattering peak in

the retro-reflection direction (hs = -h0) as the angle of

incidence h0 is increased from 0� to 45�. The results of

SIBC calculation were fitted to exact results at different

angles of incidence. We can say that in the case of p-po-

larization, the agreement between the present results and

the results obtained through exact calculations is very

good, at different angles of incidence.

At normal incidence, a more intense and acute peak in

the retro-reflection direction is seen. This peak is also

observed for the incidence angles equal to 20�, 30� and 45�.It is shown that this peak is less intensive and very wide

when the angle of incidence increases. These observations

are verified by both methods.

In Fig. 2a, b, c, and d, we noted that the bidirectional

reflectivity (BDR) calculated by SIBC method is less

intense than that presented in [6]. This observation can be

explained by the limit of validity of the present method.

In Fig. 3, we compare the results of the present method

and numerical results taken from Ref. [6]. We present the

variation of the BDR as a function of the scattering angle hsfor the scattering of s-polarized light incident normally on a

random rough surface ruled on the surface of silver. In

Fig. 3a, we observe a good agreement between the results

found by SIBC method and those given in Ref. [6]. At

normal incidence, the peak is well defined. This peak is

very intense. Figure 3b is the same as Fig. 3a, only when

the angle of incidence is 20�. Here, the peak of the BDR

calculated by SIBC method is not observed, in the

backscattering direction (hs = -20�). This can be

explained by the limit of validity of the present method and

the nature of polarization. The peak is visible at an angle of

incidence hi equal to 20�Figure 4 shows the variation of the BDR as a function of

the scattering angle hs for the scattering of s- and p-po-

larized light incident normally on a random rough metallic

surface. The surface material is the gold characterized by

its permittivity e(x) = -9.8949 ? 1.0458i. It is estimated

that the curves found by the SIBC method and those given

in ref [41] have the same allure in both cases of polariza-

tion. The peak is observed at normal incidence. This peak

found by the present method is less intense and less closed

than the peak given in Ref. [41]

Figure 5 shows curves displaying the reflectivity versus

the angle of incidence hi in both cases of polarization. The

surface material is the gold. Figure 5a and b shows the

Opt Rev

123

behavior of the reflectivity of the gold for two different

roughnesses. In Fig. 5a, we note that the value of reflec-

tivity is constant for angles of incidence less than 60�. Atlarge angles of incidence (more than 60�), the value of

reflectivity increases and tends to unity. However, in

Fig. 5b, it is seen that the value of reflectivity is constant

for angles of incidence less than 70� and for angles of

incidence more than 70� the value of reflectivity increases

and tends to unity, This difference of variation of the

reflectivity can be explained by the influence of the surface

roughness; the surface in Fig. 5a is rougher than the surface

in Fig. 5b so the value of reflectivity in Fig. 5a is less than

that in Fig. 5b. For the case of p-polarization, in Fig. 5b,

we talk about the limit of validity of the SIBC method

because for large incidence angle, the value of the reflec-

tivity exceeds unity.

Figure 6 illustrates the variation of emissivity as a

function of the ratio s/k for two different surface materials

(gold and silver) in both cases of polarization and at normal

incidence (hi = 0�). The RMS height r is fixed to 1k and

(a) (b)

(c) (d)

Fig. 2 The bidirectional

reflectivity (BDR) for the

scattering of p-polarized beam

of light from random rough

metallic surface characterized

by: r = 1.4142 lm, s = 2 lm,

k = 0.6127 lm. The surface

material is the silver with

e(x) = -17.2 ? i0.498, while

L = 25.6 lm, Ns = 300,

Np = 1024

(a) (b)Fig. 3 The bidirectional

reflectivity (BDR) for the

scattering of s-polarized beam

of light from random rough

metallic surface characterized

by: r = 1, 2 lm, s = 2 lm.

k = 0.6127 lm. The surface

material is the silver with

e(x) = -17.2 ? i0.498, while

L = 25.6 lm, Ns = 300,

Np = 1024

Opt Rev

123

the correlation length s is varied from 0 to 28k. In Fig. 6a,

we note that for 0\ s/k\ 4 the surface considered is very

rough, the intensity of emissivity increases and we have a

resonant peak, at s/k equal to 1.5. This peak is due to the

emission mediated by surface plasmon [48, 49]. For s/k[ 4, the value of emissivity is invariant when the ratio s/

k increases and tends to the value of emissivity of a flat

surface given by using Fresnel’s formula [39]. It is seen

that SIBC result show the resonance region of the material

(the gold) in both cases of polarization. In Fig. 6b, for s/k\ 4, we have a slight variation of the value of emissivity.

Unlike gold, silver does not have a resonance region. This

(a) (b)Fig. 4 The bidirectional

reflectivity (BDR) for the

scattering of s-polarized beam

of light from random rough

metallic surface characterized

by: r = 1,9 lm, s = 4.58 lm,

k = 0.633 lm. The surface

material is the gold with

e(x) = -9.8949 ? 1.0458i,

while L = 25.6 lm, Ns = 300,

Np = 1024

(a) (b)Fig. 5 The reflectance from

two different gold one-

dimensional rough surfaces in

both cases of polarization with

different roughness. The length

of each surface is L = 25.6 lm,

Ns = 300, Np = 1024

(a) (b)Fig. 6 Emissivity from two

different random rough metallic

surfaces in both cases of

polarization at normal

incidence. The length of each

surface is L = 25.6 lm,

Ns = 300, Np = 1024

Opt Rev

123

observation can be described on the basis of complex

dielectric constants [50] as real part of the dielectric per-

mittivity of material determines the resonance region

position while imaginary part determines the relative

contribution of absorption. In the case p-polarization,

emissivity decreases whereas in the case of s-polarization

emissivity is slightly increases. For s/k[ 4 the value of

emissivity tends to the value of emissivity of a flat surface.

In addition, when the ratio s/k is increased, the emissivity

tends to the value of a plane surface. Fig. 7, is the same as

Fig. 6 but at angle of incidence hi = 60�. Here, we study

the shadowing effects, and the influence of the angle of

incidence, on the behavior of emissivity. It is shown that

curves in Fig. 7a and b have the same asymptotic behavior

in both cases of polarization. For s/k\ 4, the value of

emissivity decreases at s/k equal to 2. This observation is

due to the shadowing effects however, if s/k[ 4, the value

of the emissivity is invariant and tends to the value of

emissivity of a plane surface.

6 Conclusion

A comparison between exact method and SIBC method of

random rough metallic surfaces is investigated and dis-

cussed. The effects of the angle of incidence, nature of

polarization, and the geometrical and physical parameters

on the validity of SIBC method are discussed. Backscat-

tering phenomenon for p-polarized beam of light is

observed, whereas in the case of s-polarized beam of light

is absent. However, for the surface materials of gold and

silver, the domain of validity of SIBC method is quantified

for reflection and emission in both cases of polarization.

From the result obtained, some important conclusions

are derived concerning the domains of validity of SIBC

method:

• The region of validity of SIBC method depends on the

nature of polarization, incident wavelength, direction of

incidence, and the physical parameters of surface

material.

• The domains of validity of the SIBC model are more

extensive for p-polarization than for s-polarization for

gold and silver.

• The accuracy of SIBC method is better for gold

material than for silver material. The resonance region

for gold is clear but is absent for silver.

• The use of the SIBC method is less expensive and it

requires a very short computation time.

• SIBC is suitable for dielectric and metallic surfaces.

References

1. Nieto-Vesperinas, M., Soto-Crespo, J.: ‘Monte Carlo simulations

for scattering of electromagnetic waves from perfectly conducting

random rough surfaces’. Opt. Lett. 12, 979–981 (1987)2. Maradudin, A.A., Mendez, E.R., Michel, T.: ‘Backscattering

effects in the elastic scattering of p-polarized light from a large-

amplitude randommetallic grating’. Opt. Lett. 14, 151–153 (1989)3. Saillard, M., Maystre, D.: ‘Scattering from metallic and dielectric

rough surfaces’. J. Opt. Soc. Am. A 7, 982–990 (1990)

4. Soto-Crespo, J.M., Nieto-Vesperinas, M.: Electromagnetic scat-

tering from very rough random surfaces and deep reflection

gratings. J. Opt. Soc. Am. A 6, 367–384 (1989)

5. Sanchez-Gil, J.A., Nieto-Vesperinas, M.: Light scattering from

random rough dielectric surfaces. J. Opt. Soc. Am. A 8,1270–1286 (1991)

6. Maradudin, A.A., Michel, T., McGurn, A.R., Mendez, E.R.:

‘‘Enhanced backscattering of light from a random grating’’. Ann.

Phy, 203, 255–307, (1990)7. Nieto-Vesperinas, M., Sanchez-Gil, J.A.: Light transmission from

a randomly rough dielectric diffuser: theoretical and experimental

results. Optical Lett. 15, 1261–1263 (1990)

8. Maystre, D., Saillard, M., Ingers, J.: Scattering by one or two

dimensional randomly rough surfaces. Waves Random Media 3,143–155 (1991)

(a) (b)Fig. 7 Emissivity from two

different random rough metallic

surfaces in both cases of

polarization at large incidence

(h = 60�). The length of each

surface is L = 25.6 lm,

Ns = 300, Np = 1024

Opt Rev

123

9. Ishimaru, A., Chen, J.S., Phu, P., Yoshitomi, K.: Numerical,

analytical, and experimental studies of scattering from very rough

surfaces and backscattering enhancement. Waves Random Media

1, 91–107 (1992)

10. Maradudin, A.A., Lu, J.Q., Tran, P., Wallis, R.F., Celli, V., Gu,

Z.-H., McGurn, A.R., Mendez, E.R., Michel, T., Nieto-Vesperi-

nas, M., Dainty, J.C., Sant, A.J.: Enhanced backscattering from

one- and two-dimensional random surfaces. Revista Mexicana de

Fisica 38(3), 343–397 (1992)

11. Wagner, R.L., Song, J., Chew, W.C.: Monte Carlo simulation of

electromagnetic scattering from two dimensional random rough

surfaces. IEEE Trans. Antennas Propag. 45, 235–245 (1997)

12. Colliander, A., Yla-Oijala, P.: Electromagnetic scattering from

rough surface using single integral equation and adaptive integral

method. IEEE Trans. Antennas Propag. 55, 3639–3646 (2007)

13. Heinzel, A., Boerner, V., Gombert, A., Blasi, B., Wittwer, V.,

Luther, J.: Radiation filters and emitters for the NIR based on

periodically structured metal surfaces. J. Mod. Opt. 47,2399–2419 (2001)

14. Sai, H., Yugami, H., Akiyama, Y., Kanamori, Y., Hane, K.:

Spectral control of thermal emission by periodic microstructures

in the near-infrared region. J. Opt. Soc. Am. A 18, 1471–1476(2001)

15. Marquier, F., Joulain, K., Mullet, J.P., Carminati, R., Greffet, J.-

J.: Engineering infrared emission properties of silicon in the near

field and the far field. Opt. Commun. 237, 379–388 (2004)

16. Sai, H., Kanamori, Y., Yugami, H.: High-temperature resistive

surface grating for spectral control of thermal radiation. Appl.

Phys. Lett. 82, 1685–1687 (2003)

17. Qiu, J., Zhang, W.J., Liu, L.H., Hsu, P.-F., Liu, L.J.: ‘‘Reflective

properties of randomly rough surfaces under large incidence

angles, J. Opt. Soc. Am. A. 31, 1251–1258 (2014)

18. Sai, H., Kanamori, Y., Hane, K., Yugami, H.: Numerical study on

spectral properties of tungsten one-dimensional surface-relief

gratings for spectrally selective devices. J. Opt. Soc. Am. A 22,1805–1813 (2005)

19. Bruce, N.C.: Scattering of light from surfaces with one-dimen-

sional structure calculated by the ray-tracing method. J. Opt. Soc.

Am. A 14, 1850–1858 (1997)

20. Thorsos, E.I.: The validity of the Kirchhoff approximation for

rough surface scattering using a Gaussian roughness spectrum.

J. Acoust. Soc. Am. 83, 78–92 (1988)

21. Nieto-Vesperinas M.: Scattering and diffraction in physical

optics. Wiley (1991)

22. Dimena, R.A., Buckius, R.O.: Quantifying specular approxima-

tion for angular scattering from perfectly conducting random

rough surfaces. J. Thermophys. Heat Transfer 8, 393–399 (1994)

23. Axline, R.M., Fung, A.K.: Numerical computation of scattering

from perfectly conducting random surfaces. IEEE Trans.

Antennas Propag. 26, 482–488 (1978)

24. Chan, H.L., Fung, A.K.: A numerical study of the Kirchhoff

approximation in horizontally polarized backscattering from a

random surface. Radio Sci. 13, 818–881 (1978)

25. Fung, A.K., Chen, M.F.: Numerical simulation of scattering from

simple and composite random surfaces. J. Opt. Soc. Am. A 2,2274–2284 (1985)

26. Chen, M.F., Fung, A.K.: A numerical study of the regions of

validity of the Kirchhoff and small-perturbation rough surface

scattering models. Radio Sci. 23, 163–170 (1988)

27. Tang, K., Buckius, R.O.: Regions of validity of the geometric

optics approximation for angular scattering from very rough

surfaces. Int. J. Heat Mass Transf. 40, 49–59 (1997)

28. Tang, K., Buckius, R.O.: The geometric optics approximation for

reflection from two-dimensional random rough surfaces. Int.

J. Heat Mass Transf. 41, 2037–2047 (1998)

29. Sanchez-Gil, J.A., Nieto-Vesperinas, M.: Resonance effects in

multiple light scattering from statistically rough metallic surfaces.

Phys. Rev. B. 45, 8620–8633 (1992)

30. Lochbihler, H., Depine, R.: Diffraction from highly conducting

wire gratings. Appl. Opt. 32, 3459–3465 (1993)

31. Lochbihler, H., Depine, R.: Characterization of highly conducting

wire gratings using an electromagnetic theory of diffraction. Opt.

Commun. 100, 231–239 (1993)

32. Lochbihler, H., Depine, R.: Diffraction from highly conducting

wire gratings of arbitrary cross section. J. Mod. Opt. 48,1273–1298 (1993)

33. Skigin, D.C., Depine, R.: Enhancement of antispecular orders

from metallic gratings with rectangular grooves. Optik (Jena)

101, 63–72 (1995)

34. Depine, R.A., Simon, J.M.: Diffraction grating efficiencies con-

formal mapping method for a good real conductor. J. Mod. Opt.

29, 1459–1473 (1982)

35. Depine, R.A., Simon, J.M.: Surface impedance boundary condi-

tion for metallic diffraction gratings in the optical and infrared

range. J. Mod. Opt. 30, 313–322 (1983)

36. Depine, R.A.: Perfectly conducting diffraction grating formalisms

extended to good conductors via the surface impedance boundary

condition. Appl. Opt. 26, 2348–2354 (1987)

37. Tsang, L., Chan, C.H., Pak, K., Sangani, H.: Monte Carlo sim-

ulations of large scale problems of random rough surface scat-

tering and applications to grazing incidence with the

BMIA/canonical grid method. IEEE Trans Antennas Propag.

43(8), 851–859 (1995)

38. Ghmari, F., Sassi, I., Sifaoui, M.S.: Directional hemispherical

radiative properties of random dielectric rough surfaces. Waves

Random Complex Media 15, 469–486 (2005)

39. Sassi, I., Sifaoui, M.S.: Comparison of geometric optics

approximation and integral method for reflection and transmis-

sion from micro geometrical dielectric surfaces. J. Opt. Soc. Am.

A 24, 451–462 (2007)

40. Bruce, N.C.: Calculations of the Mueller matrix for scattering of

light from two-dimensional surfaces. Waves Random Media 8,15–28 (1998)

41. Kim, M.J., Dainty, J.C., Friberg, A.T., Sant, A.J.: ‘Experimental

study of enhanced backscattering from one- and two-dimensional

random rough surfaces’. J. Opt. Soc. Am. A 7, 569–577 (1990)

42. Sassi, I., Ghmari, F., Sifaoui, M.S.: ‘‘ Effect of the material of

rough surfaces and the incident light polarization on the validity

of the surface impedance boundary condition and the geometric

optics approximation for reflection and emission. J. Opt. Soc.

Am. A 26, 480–488 (2009)

43. Carminati, R., Greffet, J.J.: Heat transfer, proceeding of 11th

IHTS. 7 August Kyongyu, Koria, 23–28 (1998)

44. Depine, R.A.: Diffraction from metallic rough surfaces. A rig-

orous electromagnetic formalism applied to the derivation of a

new surface boundary condition. Optik 79(2), 75–80 (1988)

45. Brekhovskikh, L.M.: Waves in layered media, p. 14. Academic

Press, New York (1960)

46. Stratton, J.A.: Electromagnetic Theory, 1st edn, pp. 23–131.

McGraw Hill, New York (1941)

47. Brewster, M.Q.: ‘‘Thermal radiative transfer and properties,’’

chap. 2 and 4, Wiley, New York (1992)

48. Ghmari, F., Ghbara, T., Laroche, M., Carminati, R., Greffet, J.J.:

Influence of micro roughness on emissivity. Appl. Phys. 96,2656–2664 (2004)

49. Sai, H., Kanamori, Y., Yugami, H.: Tuning of the thermal radi-

ation spectrum in the near-infrared region by metallic surface

microstructures. J. Micromech. Microeng. 15, 243–249 (2005)

50. Palik, E.D.: Handbook of Optical Constants of Solids, Academic

Press Handbook, vol. 1 (1985)

Opt Rev

123