Vol.3, No.1, 57-64 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.31008
Study of the emissivity of rough surfaces periodic using
the method of coupled waves analysis (CWA) compared
with method of geometrical optics approximation (GOA)
Taoufik Ghabara
Department of Physics-Faculty of Science and Arts at Al-Rass-Qassim University, Qassim, Kingdom of Saudi Arabia;
Received 23 June 2010; revised 25 July 2010; accepted 28 July 2010.
ABSTRACT
We present in this paper a numerical study of
the validity limit of the geometrical optics ap-
proximation compared with a differential method
which is established according to rigorous for-
malisms based on the electromagnetic theory.
The precedent studies show that this method is
adapted to the study of diffraction by periodic
rough surfaces. We determine by two methods
the emissivity of gold and tungsten for surfaces
with a rectangular or sinusoidal profile, for a
wavelength equal to 0.55 microns. The mono-
chromatic directional emissivity of these sur-
faces clearly depends on the angle of incidence,
the surface profile, height, period and the nature
of the material. We perform our calculations by
a method of coupled wave analysis (CWA) and a
geometric optics method (GOA). The latter
method is theoretically valid only when the di-
mensions of the cavities are very large compared
to the wavelength, while the CWA is theoreti-
cally correct whatever these dimensions. The
main purpose of this work is to investigate the
validity limit of GOA compared with CWA. The
obtained results for a fixed height of the grating,
allowed us to delimit the validity domain of the
optic geometrical approximation for the treated
cases. Finally, the agreement between the
emissivity calculated by the differential method
and that given on the basis of the homogeniza-
tion theory is satisfactory when the period is
much smaller than the wavelength.
Keywords: Periodic Roughness; Differential
Method (CWA); Geometric Optics Approximation
(GOA); Homogenization Theory; Emissivity.
1. INTRODUCTION
The theoretical or experimental determination of ra-
diative properties of rough surfaces is the subject of sev-
eral researches. The modelling of directional mono-
chromatic emissivity of a rough surface remain a subject
of theoretical, experimental and numerical researches
[1,2].
These parameters are involved in several application
areas ranging from the calculation of energy exchange
by radiation in the design of selective rough surfaces, in
addition to current applications on the semiconductor
industry [3] and the Solar energy [4].
In this context two methods are presented, one exact
and based on electromagnetic theory, while the second is
approximate and based on geometrical optics. Among
the exact methods, we mention the integral and differen-
tial methods.
While the integral methods are more effective to study
the scattering of electromagnetic waves by rough surfaces
[5,6], the differential methods seem best suited to solving
the same problem with periodic rough surfaces.
These methods established by the rigorous formalisms
based on electromagnetic theory are known under the
name of coupled wave analysis (coupled wave analysis
CWA) and they are first applied to planar gratings [7-10]
and also at deep profile periodic gratings dielectrics or
conductors [11-14].
In parallel, various versions of algorithms were pro-
posed. However, some of the solution algorithms are
unstable and to remedy this, several numerical algo-
rithms are developed to solve this problem for very deep
gratings [15-17].
The approximate method adopted is based on optic
geometrical approximation and it is only valid in the
case of macro-roughness [18-22]. As soon as the dimen-
sions of the roughness become comparable to or less
than the wavelength, this method can be relatively chosen.

T. Ghabara / Natural Science 3 (2011) 57-64 58
This work has established the region of validity be-
tween the geometric optics approximation (GOA) and
differential method (CWA) especially in the cases of
rectangular and sinusoidal surfaces with a conductivity
(gold and tungsten) lying in the visible. We have ana-
lysed the different physical phenomena depending on the
period and the angles of incidence. We also study the
radiative of the rough surface when its period is very
small compared to the wavelength. The wavy portion is
equivalent in these conditions to a layering of effective
indices determined using the effective medium theory
[23-25].
2. DERIVATION OF THE EMISSIVITY
USING DIFFERENTIAL METHOD
(CWA)
2.1. Geometry
In this study, we have considered a grating with rec-
tangular or sinusoidal grooves. An electromagnetic wave
obliquely incident upon the grating produces both re-
flected and transmitted waves, as it is shown in the Fig-
ure 1. Region 1 is a homogeneous dielectric with a rela-
tive permittivity of 1

. Likewise, region 3 is homoge-
neous with a complex permittivity 3

.
Region 2 (the grating region) consists of periodic dis-
tribution of both types of materiel. In this paper, for
simplicity, we have assumed that the incident light has
transverse electric (TE) polarization.
2.2. Theory of Differential Method
In the present analysis, the differential method CWA
(coupled Wave Analysis) is adapted to the exact elec-
tromagnetic boundary value problem associated with
dielectric and metallic miro-rough periodic surfaces
[13,14].
We consider an incident plane wave of wave vector,
located in the plane (xOz), in the case of TE polarization.
The problem is to determine the amplitudes of reflected
Reflected
waves
Incident
wave
Region 1
Region 2
Region 3
h
Transmitted
wave
Thin la
y
ers
Z
X
Figure 1. Surface with rectangular grooves.
and transmitted fields. The total electric field in Region
1 is the sum of the incident waves and reflected waves.
The amplitude of this field is given by:
 
11 1
expexp ,
ii
i
EjkrRjkr



 




 (1)
k1 is the incident wave vector; Ri is the ith amplitude
of the reflected wave in region 1. The diffracted field in
region 3 is given by:


33
exp ,
ii
i
ET jkrhz





 (2)
where i T is the amplitude of the ith transmitted wave in
region 3 in this method, a surface relief grating (region 2)
is divided in M thin layers similar to planar grating per-
pendicular to the axis (Oz) (Figure 1), then applying the
method of coupled waves at each layer [10]. The permit-
tivity of the kth layer is a periodic function by (Ox),
who’s Fourier series expansion can be expressed by:



,
1
, exp1,
lk
kk
x
zjkx 


 (3)
where k is the grating vector (2kd

),

,lk

are the
complex coefficients of Fourier series expansion. The
field in each sub-layer is given by:

2, ,,
()exp ,
kik ik
i
ESzjr 





(4)
where ,ik
S is the amplitude of the diffracted field of
order i and ,ik


is the diffracted wave vector in the
interior of the kth layer.
In region 2 we have introduced the fundamental wave
equation in TE polarization:

22
2, 2,
,0
kkkk
EkxzE 


(5)
By replacing the field expansions (3) and (4) in the
wave equation, we have obtained a differential system
with constant coefficients according to each thin grating
k:






21/2
,, 22 2
2, 1
2
22
,,,
2sin
() 0
ik ik
k
ikpki pk
ip
dS zdS z
jk kdz
dz
Kim iSzkS





 

(6)
where:


1/2
1
2sinmd

 
2.3. Method Resolution
The transformation:
 
1, ,,ik ik
SzSz and


2, ,ik
Sz


,ik
dSz dz, leads to the linear system of differential
equations of first order, is written in matrix form as fol-
lows:

T. Ghabara / Natural Science 3 (2011) 57-64
5959
  
,, ,,
'lpk lqk SbS



(7)
For l = 1,2 and 'SdSdz.
The matrix coefficients are given in [13] (p = 1, N et q
= 1, N). A solution of system (7) is given by:
2
',',', ',',
'1
exp( )
N
pkqk pqkqk
q
SCW z 


 (8)
where ',qk

is the eigenvalues and ', ',
p
qk
W is the ei-
genvectors of the matrix. ',qk
C are the constants to be
determined, first using the boundary conditions.
2.4. Derivation of the Emissivity Using
Differential Method (CWA)
For solving the problem without numerical difficulties,
we have adopted the stable algorithm presented by M.G
Moharam [14] for transverse electric polarization (TE).
When the reflected Ri and transmitted Ti field com-
plexes amplitudes are known, the diffraction efficiencies
(ratio of diffracted intensity to input intensity) may be
directly determined. Then the diffraction efficiencies in
region 1 and 3 are:

*
1110
Re
ii ii
DEkzkzRR




 (9)

*
3310
Re
ii ii
DEkzkzT T




 (10)
The emissivity is given by the following relation [26]:

1
0
1i
i
DE




 (11)
3. DERIVATION OF THE EMISSIVITY
USING GEOMETRICAL OPTICS
APPROXIMATION (GOA)
The method of geometrical optics is one of approxi-
mate methods of calculating the emissivity rough sur-
faces. It is based on the concept of light ray, and we use
the notions of classical geometry or analytical related to
Snell's law to determine the path of the incident beam
inside the cavity. We adopt the approach that the princi-
ple is to consider a beam of parallel rays incident on the
cavity at an angle of incidence and to determine the part
absorbed [31-32].
The emissivity derived from geometrical optics is
equal to:

1
,1 ()
j
n
s
i
i
X





 (12)
where i

designates the local reflection angle at the
interaction point Si (Figure 2) and

i 

 is the Fres-
nel reflection factor.
In the case of sinusoidal cavities (Figure 2), the direc-
tional emissivity is simply the average of all positions
j
s
X
, and can be written as:


0
1,j
d
s
X
dX
d

   (13)
where d is the grating period.
For rectangular cavities, determining the number of
impact points inside the cavity and the local angles of
incidence or reflection is done using a geometric study
to show that the number of reflections
j
N inside the
cavity is related to the integer part of quantity:
2tan21, ,
2
j
j
s
s
X
hd
X
d
dd


 


The directional emissivity is given by:
 
1
12
j
N


 



 
 


(14)
4. CONCEPT OF HOMOGENIZATION
For gratings with a period much smaller than the
wavelength, the roughness essentially behaves as a tran-
sition layer with a gradient of the optical index. The
emissivity is given by the following relation [26]:

2
1r

  (15)
where r is the ratio of complex amplitude of reflected
field and the complex amplitude of the incident field.
5. RESULTS AND DISCUSSIONS
We present results of our numerical calculations of the
emissivity of gratings with a rectangular or sinusoidal
Figure 2. Reflections successive associated with an incident
ray of the cavity sinusoidal. D1: incident ray; Si: interaction
points; θ incidence angle; φ1: local reflection angle.

T. Ghabara / Natural Science 3 (2011) 57-64 60
grooves for height h and period d. We associate these
surfaces defined by the angle: tan 2hd

. Note that
when d

varies from 0.05 to 12.8 angle,

varies
from 75.96˚ to 0.89˚ for 0.1h

 and from 88.56˚ to
8.88˚ for 1h

.
The grooves are composed of materials of gold (Au)
and tungsten (W), of respective refractive indexes nu =
0.48 + i2.45 and nw = 3.5 + i2.73 corresponding to a
wavelength equal 0.55m

.
5.1. Rectangular Surfaces
For an angle of incidence equal to 1˚, it follows an
agreement between the differential method (CWA) and
the geometric method (GOA) for gold and tungsten in
the following cases: From the ratio d

equal to 1.6
when the height h of the grating is equal 0.1

(Figures
3(a,b)), and ratio d

above 3.2 when h is equal 1

(Figures 3(c,d)). These two limits can be translated re-
spectively by angles

lower to 7.12˚ and 32˚.
For height h equal 0.1

, the asymptotic limit of
emissivity curves, reflecting the agreement between the
CWA and GOA corresponds to the values 0.21 and 0.51
in the case of gold and tungsten respectively [26]. These
limits are those of the emissivity of the smooth surface
in the direction 1, calculated using the Fresnel formulas
[26-30].
In these conditions

0.1 ,1h  
, the reflection is
simple at any point in the grating; the agreement be-
tween the two methods is possible only for a period d
exceeding 1.6

. It is thus clear that the validity of
GOA is not exclusively related to the number of reflec-
tions within the groove. However, it is remarkably to
note that the GOA is valid for grating for height equal
110

and period equal

. For the same angle of in-
cidence 1˚, and the same reports d

from 0.05 to 12.08,
(a)
(b)
(c)
(d)
Figure 3. Comparison of differential method (CWA) with
geometric optics approximation (GOA) for the directional
monochromatic emissivity  d


 of rectangular surfaces.

T. Ghabara / Natural Science 3 (2011) 57-64
6161
we clearly remark that the validity domain of GOA in-
creases dramatically when the height increased from
0.1

to 1

. In terms of angle

characteristic of the
cavity the domain of validity is from [0˚, 7˚] to [0˚, 32˚].
In our study, it is important to note that for a groove
height 1

, it suffices that the period exceeds 3.2

for
the GOA to become valid. In the case of figures (Fig-
ures 3(c,d)), reflection on the inside of the groove is
easy, especially since the direction of incidence very
close to normal there is no shadow effect.
5.2. Sinusoidal Surfaces
Emissivity curves

d


 , for both h and

fixed,
calculated by the two methods (Figures 4(a,b)) to de-
termine the value of the ratio limit (d

)lim from which
the GOA is valid. We associate the validity domain of
GOA to the phenomenon of multiple reflection inside
the cavity and that of the shadow effect due to the sur-
face [27,28].
This effect is negligible for directions of incidence
around normal (

= 1˚) (Figure 4(a)) because to con-
sider only simple reflection. For directions (

= 60˚)
(Figure 4(b)), the shading effect is enhanced and this
effect is also associated with the phenomenon of multi-
ple reflections.
5.3. The Range of Validity of the GOA
We consider the previous study in the case of gold for
heights h and for incidence angles from 10˚ to 80˚ with a
step of 10˚. In this report limit

lim
d  is associated,
for h fixed, the limiting slope (2hd) lim. Figure 5
shows curves the limiting slope versus the cosine of the
angle of incidence or emission

, for a fixed height h.
This curve defines two regions. For the first, situated
(a)
(b)
Figure 4. Comparison of differential method (CWA) with
geometric optics approximation (GOA) for the directional
monochromatic emissivity

d


 of sinusoidal surfaces.
Figure 5. Differential method of sinusoidal surfaces domain
plot with region validity for the geometric optics approxima-
tion in terms of limiting slope

lim
2hd .
above this curve, using the CWA or another accurate
method is necessary since GOA is not valid. This is very
satisfactory at all points of the second region located
below the same curve.
By adopting the previous approach, for h

and

fixed, by varying the ratio d

, we first determine the
limits of validity of GOA compared with CWA respec-
tively, for incidence angles 1 and 60˚ and reports h

going up to value 10 with a step equal to 2. We include
the points determined from the curves of Figure 5. Thus,
Figure 6 gives a significant idea about the range of the
geometrical optics approximation validity. In fact under
the cloud of points in this figure the approximation of

T. Ghabara / Natural Science 3 (2011) 57-64 62
geometrical optics is valid, so that above the same cloud
the use of an exact method is necessary.
6. HOMOGENIZATION REGIME
We analyze the behaviour of emissivity when the pe-
riod is very small compared to the wavelength. Under
these conditions the grating is equivalent to a superposi-
tion of layers of effective indices determined using the
effective medium theory [26]. We compare the emissivi-
ties calculated by the homogenization process, for d

equal to 0.05, in the cases discussed above with that
calculated by the CWA and GOA. We see a very good
agreement with the differential method in all these cases,
however no agreement with the method of geometrical
optics can be reported according to what we should ex-
pect (Tables 1,2).
Figure 6. Ratio

lim
2hd as function of


cosh

 .
Table 1. (a) and (b): Comparison between the emissivity cal-
culated by the homogenization process and those calculated by
the methods of differential and geometrical optics for d

equal to 0.05, in the case of TE polarization. Case of gold and
tungsten – rectangular surfaces.
(a)
h θ CWA (Gold) Homogenization (Gold) GOA (Gold)
0.1λ 1˚ 0.2612 0.2681 0.2574
0.1λ 60˚ 0.1313 0.1346 0.5399
1λ 1˚ 0.3066 0.3082 0.2589
1λ 60˚ 0.1497 0.1505 0.5409
(b)
h θ CWA
(Tungsten)
Homogenization
(Tungsten)
GOA
(Tungsten)
0.1λ 1˚ 0.6381 0.6443 0.4261
0.1λ 60˚ 0.3942 0.3997 0.5409
1λ 1˚ 0.6281 0.6320 0.4284
1λ 60˚ 0.3880 0.3913 0.5409
Table 2. (a) and (b): Comparison between the emissivity cal-
culated by the homogenization process and those calculated by
the methods of differential and geometrical optics for d

equal to 0.05, in the case of TE polarization. Case of gold and
tungsten – rectangular surfaces.
(a)
h θ CWA (Gold)Homogenization (Gold) GOA (Gold)
0.1λ1˚0.2509 0.2520 0.4712
0.1λ60˚0.1267 0.1273 0.5846
1λ 1˚0.4586 0.4620 0.7650
1λ 60˚0.2156 0.2177 0.8547
(b)
h θ CWA
(Tungsten)
Homogenization
(Tungsten)
GOA
(Tungsten)
0.1λ1˚ 0.6396 0.6449 0.7624
0.1λ60˚0.3953 0.3995 0.6747
1λ 1˚ 0.8228 0.8201 0.9319
1λ 60˚0.5670 0.5754 0.9351
7. CONCLUSIONS
In this paper, we study the validity limits of the of
geometrical optics approximation compared with a dif-
ferential method. We determine by both methods the
emissivity for rectangular and sinusoidal surfaces of
gold and tungsten, when the wavelength is equal to 0.55
microns.
The results obtained by the exploitation of the codes
elaborated for the three methods and for TE polarization
are validated. The obtained results and the presented
interpretations lead to the following conclusions:
In case of an angle of incidence equal to 1˚ and a
height of roughness equal 0.1

, we find an agreement
between the differential method and the approximation
of geometrical optics as well as for gold and tungsten for
a period greater than 1.6

.
For the same incidence angle 1˚, and the same reports
d

, we conclude that the validity domain of GOA in-
creases when the height of the roughness increases from
0.1

to 1

.
In the case of an angle of incidence equal to 60˚ and
for both materials, the agreement between the two
methods is obtained as soon as the ratio d

becomes
greater than 0.8.
The simple reflection condition for the GOA is nec-
essary but not sufficient.
For 1h


the shadowing phenomenon does not
prohibit the validity of the geometric method.
The emissivities obtained by the homogenization
method are in perfect agreement with those calculated by
the differential method when the period of the profiles is
very small compared to the wavelength.

T. Ghabara / Natural Science 3 (2011) 57-64
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