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j.infrared.2018.07.018

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Infrared Physics and Technology 93 (2018) 25–33
Contents lists available at ScienceDirect
Infrared Physics & Technology
journal homepage: www.elsevier.com/locate/infrared
Regular article
Scattering properties of solid rough surface of nickel skeleton
⁎
T
Bo Liu, Xinlin Xia , Chuang Sun
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
A R T I C LE I N FO
A B S T R A C T
Keywords:
Metal foam
Micromorphology of solid skeleton
Scattering properties
FDTD
The micromorphology of solid skeleton makes a significant influence on the radiative transfer in metal foams.
The modeling of complex morphology of rough surface and the prediction of bidirectional reflectance distribution function (BRDF) are the key issues for the radiative calculation of metal foam. In this work, a modeling
method of analyzing and simplifying the complex rough surface using the SEM figures of metal foam’s skeleton is
introduced. This method is also used to predict the scattering properties of nickel foam’s skeleton. Meanwhile,
the hemispherical structures are used to replace the real convex structures on skeleton’s surface. These hemispherical convex are located on the surface randomly, and the radius of adapted hemispherical convex respect
the distribution function extracted from SEM statistic data. Furthermore, the FDTD method is adapted to predict
the reflectivity and BRDF of simplified rough surfaces models. The results show that the reflectivity is affected by
the size parameter obviously, while it’s changed slightly by different incident angles. As for the BRDF of skeleton’s rough surfaces, the value of main peaks increase obviously and the scopes of main peaks decrease with
the size parameter increasing. The sub-peaks locate differently surrounding the main peaks, and then fuse
gradually with the increasing size parameter. The sub-peaks vanish completely at the size parameter up to 6.
When it comes to the hemispherical space, the reflection vary from anisotropic reflection to isotropic reflection
except the mirror direction. At a larger size parameter, the reflection of rough surfaces can be simplified as the
combination of mirror reflection and diffuse reflection. While at a small size parameter, the BRDF of rough
surface need to be calculated accurately to reduce the error in the prediction of radiative transfer.
1. Introduction
Metal foam is an attracting wide focus on thermal applications, such
as high-temperature exchangers [1,2], solar receivers [3,4] and so on.
Radiative transfer is vital for the heat and mass transfer of metal foam.
The structure inside metal foam has great influence on the radiative
transfer. There are lots of excellent works for the research of the radiative transfer of metal foam. Rousseau [5] made the numerical prediction of the radiative behavior of metallic foams with the Monte Carlo
Ray Tracing (MCRT) program. Li made the prediction of the spectral
reflection behaviors of high-porosity metal foam sheets, bidirectional
reflectance distribution function and directional-hemispherical reflectivity with the Monte Carlo ray tracing method [6], and a predictive
relationship was established that links the homogenized apparent
emissivity to the open porosity and the intrinsic emissivity of solid
struts [7]. The modeling of the infrared surface temperature of opencell metallic foam with the discrete-scale approach (DSA) and continuous-scale approach based on Backward Monte Carlo simulation was
investigated by Li and Xia [8]. However, in most researches about the
radiative transfer in metal foam, the radiative properties of skeleton’s
⁎
Corresponding author.
E-mail address: xiaxl@hit.edu.cn (X. Xia).
https://doi.org/10.1016/j.infrared.2018.07.018
Received 17 April 2018; Accepted 11 July 2018
1350-4495/ © 2018 Elsevier B.V. All rights reserved.
surfaces are usually from assumptions and empirical formulas. Actually,
due to the complex morphology on the rough surface of skeleton, the
radiative properties of metal foam’s skeleton are usually complex and
cannot be described with uniform functions.
For getting a better prediction of radiative transfer in metal foam,
the radiative properties of skeleton’s rough surface need to be calculated accurately. The extreme complex morphology of skeleton’s surface produces large error between the actual anisotropic reflection and
the regular expression of functions, and the complex morphology of
skeleton’s surface makes the radiative transfer in metal foam hard to
predict. So the radiative properties of rough surface are the key issues
for further study. For predicting the radiative behavior of rough surfaces, plenty of outstanding works have been fulfilled. Parviainen [9]
studied the light scattering from self-affine homogeneous isotropic
random rough surfaces by using the ray-optics. Zhang [10] applied the
updated conventional geometrical optics ray-tracing method by considering the effect of the interference to study the radiative properties of
a one-dimensional random rough surface and compared with the results
by applied the FDTD method. Feng [11] measured the polarized BRDF
of surfaces with an experiment and verified a hybrid model of polarized
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
BRDF for rough surfaces. Wang [12] established a five-parameter BRDF
model of rough surface and made an experimental verification within
infrared band. Kang [13] modeled the micro-scaled rough surface respecting the Gaussian distribution and predicted the radiative properties with the FDTD method. Qi [14] analyzed the BRDF of fractal rough
surface by applying the numerical calculation. The spectral reflectivity
of a periodic surface in the thermal radiation wavelength range was
predicted by Mendeleyev [15] and the influence of orientation of rough
grooves were discussed. However, almost all the present works are
focus on the rough surfaces of bulk material. What’s more, the surfaces
are usually modeled by mathematical expression, such as the Gaussian
surfaces and the fractal surfaces. However, there are obvious differences between the bulk material and the skeleton of metal foam due to
the various fabrication process. Few studies consider the real morphology of skeleton’s surface, which can’t be described with simple
mathematical formulas. The surface of skeletons are usually complex
since the fabricating technique [16–18] and the real morphology need
to be analyzed by applying advanced technique. Coquard [19] considered the influence of skeleton’s rough surface on the radiative
properties of metal foam and predicted the radiative transfer with
geometric optics since the characteristic sizes of the cells are almost
always much greater than infrared radiation wavelengths, and the sizes
of structures on the skeleton surface were still in the range of geometric
optics. However, the application of geometric optics is limited by the
size of structures. The Maxwell equations and other electromagnetic
theories [20] are needed to study the radiative properties of skeleton
rough surfaces when the characteristic sizes of structure are similar to
the wavelength. The prediction of radiative properties of skeleton’s
surfaces inside metal foam hasn’t attracted further attention.
For getting the accurate radiative properties of micro-scaled structures, the numerical techniques should be adapted. The finite-difference
time-domain (FDTD) method [21] is one of widely used methods for the
radiative transfer of micro-scaled structures. Qiu [22] made an analysis
of infrared radiative properties of one dimensional periodic aluminum
surfaces with various micro-scale triangular gratings using the FDTD
method. Chen [23] applied a computational model based on the finitedifference time-domain method and the Wiener Chaos Expansion
(WCE) method to calculate the near-field radiative heat transfer between two plates with Gaussian type rough surfaces. Wang [24] studied
the absorption characteristics of plasmonic metamaterials with an array
of nanoshells with the FDTD method. The reliability of the FDTD
method has been proved by plenty of works.
In this work, by applying the SEM technique, a method is introduced
for building the simplified model of rough surface of solid skeleton. And
the skeleton’s surface of Ni foam is analyzed by applying the method.
The FDTD method is adapted to get the reflectivity and BRDF of skeleton’s surface with various parameters. The results are compared and
discussed. This work can be benefit to the analysis of radiative properties for the radiative transfer in whole metal foam. The model adapted
is introduced in Section 2. In Section 3, the numerical method and
calculation condition are presented. The results and analyses are discussed in Section 4 and the conclusions are summarized in the following section.
Fig. 1. The surface of electrolytic Ni foam’s skeleton from SEM scanning.
method is introduced for modeling the simplified surfaces, and then the
method is adapted to model the skeleton’s surfaces of electrolytic Ni
foam. The operations of the introduced method are shown as following.
(a) Firstly, the figures of skeleton’s surface such as Fig. 1 are got by
applying the SEM technique. The characteristics of the structures on
the skeleton surfaces are observed and summarized. The regular
structures with common shapes are adapted to simplify the microscale complex structures on the rough surfaces. The adapted simplified structures are approximate to the micro-scaled structures on
the surfaces.
(b) Secondly, the technique for analyzing the scanning pictures is used
for getting the statistic data of surface’s micro-scaled structures. In
this process, the sizes, amounts and other characteristics are
counted and analyzed. The statistic data is used to model the microscaled structures, and the modeling of simplified rough surfaces is
carried out later.
(c) Furthermore, the simplified surfaces are modeled with the chosen
regular structures. The sizes and locations of adapted simplified
structures are from the statistic data above. The modeled rough
surfaces are approximate to the scanning pictures of skeleton and
own the statistical characteristics.
(d) At last, the radiative properties of modeled rough surfaces can be
predicted after calculating. The predicted radiative parameters are
compared and analyzed. The predicted properties are the basis of
further research about the radiative transfer of metal foam.
For better understanding of the introduced method and further researches, the rough surfaces of electrolytic metal Ni foam’s skeleton are
modeled by applying the method above. The details and results are
shown in following sections in this work.
After observing and analyzing Fig. 1, the skeleton’s surface is rough
and made up of abundant random convex structures. The random
micro-scaled structures are similar to part of sphere with different radius. For simulating the real surfaces, the incomplete spheres are
adapted to instead the convex structures on the surface. The adapted
approximate hemispherical structure is shown in Fig. 2.
The hemispherical structure is characterized by two features, including the bottom radius r and the height h as shown in Fig. 3.
For better characterizing the surface’s morphology, the technique
for analysis of scanning figure is used to analyze the scanning picture
shown in Fig. 1, the bottom radius of different convex structures and
the distances between adjacent convex structures are measured and
counted. For 500 convex structures are measured in this work, the
statistic data of radius are shown as Fig. 4 followed.
In Fig. 4, r is the bottom radius of convex structure, and N is the
2. Model
For the radiative transfer in metal foam, the prediction for radiative
properties of skeleton’s surface is a significant issue. The surfaces of
skeletons are usually complex in morphology, which affect the radiative
parameters of the whole foam. The structures on the skeleton’s surface
are usually wavelength-scaled as shown in Fig. 1.
Fig. 1 is the scanning picture of skeleton of electrolytic Ni foam with
the SEM technique. The morphology of metal foam’s skeleton is complex and difficult in constructing the real three-dimensional structure.
Therefore, simplified rough surfaces are needed and the methods of
simplifying the surfaces deserve further researches. In this work, a
26
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
Fig. 2. The incomplete sphere structure adapted to model the convex.
number of convex that has the corresponding radius r. From Fig. 4, the
distribution of radius satisfies the Gaussian distribution. The least
square method is used to fit the statistic data and the fitting curve is
shown in Fig. 4. The occupying proportion P of convex structures with
corresponding radius r can be expressed by the equation as followed
P = 0.1804e (−((r − 0.975)/0.2954)
2)
(1)
From Eq. (1), the distribution of convex’ radius can be described by
the Gaussian distribution function with mean value μ = 0.975 μm and
standard deviation σ = 0.209.
The function of occupying proportion shown in Eq. (1) is used to
characterize the distribution of convex’ bottom radius on the rough
surfaces discussed here. For each convex structure, the random bottom
radius satisfies the function of occupying proportion, and the ratio of
height to radius r/h respects the uniform distribution from 0.1 to 0.9.
According to the statistic results of the SEM figure 4, 100 convex
structures are located on the surface with 25 × 25 μm. The modeled
surface with the method above is shown in Fig. 5.
The size parameter χ shown in Eq. (2) is used to characterize the
surfaces roughness. The λ is the wavelength of incident light, and r is
the mean value of bottom radius.
χ=
2πr
λ
Fig. 4. The statistic data of convex structures’ radius on the skeleton’s surface.
(2)
In this work, the size parameters χ of discussed rough surfaces are 1,
3 and 6. When the sizes of convex structures are similar to the incident
wavelength, the computational electromagnetic is needed for the prediction of radiative properties due to the fact that the geometrical optics
is limited by the characteristic size.
3. Method
The recent numerical calculating methods adapted widely usually
need abundant computation resource and these method like the FDTD
method, the method of moments and others are time consuming. Thus
Fig. 5. The modeled surface with the method introduced above.
Fig. 3. The parameters of modeled convex by using the incomplete sphere.
27
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
N (ω) = ∫ Js (ω) exp(jkr ′·r ) ds′
the domain of computation is always limited in small area by applying
numerical technique to solve the Maxwell equations. The work here for
the prediction of rough surfaces’ radiative parameters is based on the
FDTD method. For the non-magnetic material, the Maxwell equations
and constitute equation are as followed:
∂D
∇×H=
∂t
∇ × E = −μ0
S
L (ω) = ∫ Ms (ω) exp(jkr ′·r ) ds′
S
where j = −1 , k is the wave number, r is the unit vector from the
origin point to the far field and r′ is the vector from the origin point to
the source point.
The electric field in the far field can be described as the following
Eq. (11):
(3)
∂H
∂t
(4)
Eθ = j exp(−jKR)(−ηNθ + Lϕ )/(2λR)
Eϕ = j exp(−jKR)(−ηNϕ + Lθ )/(2λR)
(5)
D = ε0 εr E
∫ Sds = ∫ 12 Re[E × H ∗] ds
a
a
(6)
where H* is the complex conjugate of the magnetic field vector, s is the
area of the surface. In this work, the reflectivity and BRDF are discussed. The reflectivity R is defined by
R=
Pr
Pi
(7)
where Pr is the power of reflected light and Pi is the power of incident
light. The BRDF in this work are defined by equations as following
BRDF (θr , φr ) =
dPr (θr , φr )
Pi (θi , φi ) cos(θr ) dΩr (θr , φr )
4. Results
(8)
where θi, φi are the zenith angle and azimuth angle of incident light,
and the θr, φr denote the zenith angle and azimuth angle of reflected
light. dΩi and dΩr are the detector solid angles of incident light and
reflected light. The dΩ is shown in Fig. 6.
For getting the BRDF of modeled rough surface of metal foam, the
transformation from radiation in near-field to far-field is accomplished
with the equivalence principle.
For getting the radiative energy in far field, a closed imaginary
surface is located in the computation domain, and has the normal
vector n. The equivalent electric current Js and magnetic current Ms can
be get from the following equations.
Js (ω) = n × H (ω)
Ms (ω) = −n × E (ω)
(11)
where R is the distance from the source point, η is the wave impedance
in free space. After getting the electric field in far field, the radiative
power and BRDF can be calculated. The radiative parameters can be
predicted after the calculation above.
In this work, the incident light is non-polarization and the wavelengths are in the near infrared band. The boundary of calculating
domain is period for the computation with the FDTD method.
For verifying the accuracy and validity of the FDTD method adapted
in this work, the absorptivity of tungsten grating calculated by the
RCWA in reference [25] are calculated again by applying the adapted
FDTD method in this work. The grating has periodic length of 1.6 μm,
height of 0.2 μm and width of 1.28 μm. The results predicted by the
FDTD method here are compared with the calculated results in reference [25] and depicted in Fig. 7. From the comparison in Fig. 7, the
FDTD method used in this work is reliable for the prediction of radiative
properties of discussed structures.
where H, E and D represent the magnetic field, the electric field and the
electric displacement vector. μ0 is the permeability of vacuum, the ε0 is
the permeability of vacuum. εr is the relative permittivity. The timeaveraged power flow across a surface is defined by
p=
(10)
In this work, a method is introduced for analyzing the rough surfaces of skeleton and modeling the simplified surface. The method is
carried out according to the statistic characteristics of micro-scaled
structures on the metal foams’ skeleton. Then, the rough surfaces of Ni
foam’s skeleton is modeled by the approach. The modeled rough surfaces are made up of various random hemispherical convex structures
shown in Section 2. The radiative properties including the reflectivity
and BRDF are predicted with the FDTD method. The complex refraction
n + ik of metal Ni are shown in Fig. 8 [26]. In the following prediction
for the radiative properties of modeled skeleton’s surfaces, five different
surfaces with same parameters are produced by the modeling method
introduced above. The five surfaces are adapted to model the skeleton
surface. The radiative parameters of the surfaces are calculated. The
(9)
where ω is the frequency of electromagnetic wave. The potential
functions can be defined as Eq. (10).
Fig. 7. The comparison between the results from FDTD method and reference
[25].
Fig. 6. The hemisphere space above the calculated surface.
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Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
whole hemispherical space above the rough surfaces. The numbers of
calculated positions for the prediction of BRDF are about 160,000 for
different cases, and these positions distribute over the discrete zenith
angle and azimuth angle. The figures in following sections are depicted
with points dropped since the calculated points are too dense.
4.1. The reflectivity
The reflectivity of rough surfaces of Ni foam’s skeleton discussed
above are shown in Fig. 9. The size parameters of modeled surfaces are
1, 3 and 6 with the incident angles are 0°, 30° and 60°.
The reflectivity of different cases with different incident angles and
size parameters are shown in Fig. 9. From Fig. 9, the reflectivity increases with decreasing size parameter. The reflectivity of different
incident angles are similar for a certain size parameter. The variety is
the same as the various complex refraction shown in Fig. 8. For larger
incident angle of 60°, the reflectivity decrease slightly for all the size
parameters. The reflectivity is affected by the size parameter obviously
and change slightly with different incident angles.
Fig. 8. The complex refraction n + ik of metal Ni.
4.2. The BRDF
The BRDF of rough surface is scattered over the whole hemisphere
space. The BRDF in hemisphere space shows the distribution of reflected energy by the rough surface. In this work, the top views are
shown in following discussions which are more convenient to be observed and analyzed in case that the distribution in whole space is hard
to observe completely. The whole hemisphere space is shown in
Fig. 10(a). What’s more, the BRDF in the incident plane shown in
Fig. 10(b) is extracted for further analysis. The incident light and reflected light are shown in Fig. 10(b). In this plane, the +y direction
correspond to the incident angle θi range from 0° to 90°, while the reflected angle θr is from −90° to 0°.
4.2.1. The rough surface of χ = 1
The rough surface’s BRDF in hemispherical space with size parameter χ = 1 are shown in Fig. 11. From Fig. 11, there is one main peak
of BRDF in the mirror direction of incident light. And several sub-peaks
exit surrounding the main peak. In other directions, the values of BRDF
are very small compared with the main peaks and sub-peaks, and the
BRDF of other directions are anisotropic obviously. The BRDF in the
incident plane are shown in Fig. 11(d), the values of main peaks are
similar under different incident angles. Furthermore, the mirror reflections take the domination, and own most of reflected power. From
Fig. 11(d), for the bigger incident angle, the location of main peak
deviates slightly from the direction of mirror direction.
Fig. 9. The reflectivity of metal Ni foam’s skeleton rough surface with various
parameters.
mean results of the five different surfaces are used to characterize the
properties of surfaces for removing the error produced by randomicity
from the process of modeling surfaces.
In the following sections, the predicted BRDF are scattered over the
Fig. 10. The extracted incident surface for further analysis.
29
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
Fig. 11. The BRDF in hemispherical space and incident surface of rough surface with χ = 1 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the
incident angle θi is 60°; (d) the BRDF extracted from the incident plane.
4.2.2. The rough surface of χ = 3
The rough surface’s BRDF in hemispherical space with size parameter χ = 3 are shown in Fig. 12, the BRDF in incident plane are shown
in Fig. 12(d). As same as the case of χ = 1, there are main peaks in the
mirror direction for incident angle are 0° and 30°, but the main peak
deviates from the mirror location slightly for the larger incident angle
of 60°. The main peaks own most reflected power, too. However, there
are some differences for bigger size parameter, firstly, sub-peaks in
different directions fuse gradually, which causes the sub-peaks become
obscure. Furthermore, the difference between the values of main peaks
30
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
Fig. 12. The BRDF in hemispherical space and incident surface of rough surface with χ = 3 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the
incident angle θi is 60°; (d) the BRDF extracted from the incident plane.
31
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
Fig. 13. The BRDF in hemispherical space and incident surface of rough surface with χ = 6 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the
incident angle θi is 60°; (d) the BRDF extracted from the incident plane.
4.2.3. The rough surface of χ = 6
The rough surface’s BRDF in hemispherical space with size parameter χ = 6 are shown in Fig. 13. Compared with the cases discussed
above, the varieties of BRDF are similar. First of all, for largest size
parameter of χ = 6, the reflections in mirror directions become more
ever-dominant and concentrated. The scopes of main peaks are narrowed further. The values of main peaks are increased and the differences between different incident angles are enhanced further. Secondly,
under different incident angles are enlarged, and the values of main
peaks increase substantially while the scopes of the main peaks decreases obviously. The reflected power are concentrated further near
the mirror directions. What’s more, the isotropy in other directions are
enhanced. The reflections in other directions except the mirror direction are turning to the diffuse gradually.
32
Infrared Physics and Technology 93 (2018) 25–33
B. Liu et al.
reflection. However, when the size parameters are small, the BRDF of
rough surfaces need to be calculated accurately to reduce the error in
the prediction of radiative transfer. The method introduced here is a
choice for modeling the skeletons’ rough surfaces for further research.
The results of BRDF distributions of discussed surfaces that are adapted
to model the rough surfaces of Ni metal foam’s skeletons can be benefit
to the radiative transfer in metal foam further and the applications of
foam materials for later studies.
the sub-peaks in different directions fuse completely. The sub-peaks
develop to the whole region surrounding the main peaks. Thirdly, the
reflections in other directions are almost isotropy completely. This part
of reflected power can be treated as the diffuse reflection for the prediction of radiative transfer.
Take all the conditions considered, the BRDF of modeled rough
surfaces vary with increasing size parameters and incident angles. (1)
There are main peaks near the mirror direction of incident light that
own most of reflected power for all the cases. For a bigger incident
angle, slight deviation of the location appears. With the size parameter
increasing, the values of main peaks increase obviously and the scopes
of main peaks decrease. (2) For smaller size parameter, the sub-peaks
exit in different locations surrounding the main peaks. These sub-peaks
fuse gradually with increasing size parameter. When the size parameter
become 6, the sub-peaks vanish completely and become whole regions
with isotropic reflection surrounding the main peaks. (3) For other
directions expect the mirror directions in the hemispherical space, the
reflections vary from anisotropic reflection to isotropic reflection. For
the largest size parameter discussed here, the reflections in other directions can be treated as the diffuse reflection. For larger size parameters, the reflections of rough surfaces can be simplified into the
combination of mirror reflection and diffuse reflection. However, when
the size parameters are small, the BRDF of rough surfaces need to be
calculated accurately to reduce the error in the prediction of radiative
transfer.
Acknowledgement
The work is supported by the National Natural Science Foundation
of China (No. 51536001).
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in the
online version, at https://doi.org/10.1016/j.infrared.2018.07.018.
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5. Conclusion
The radiative transfer progress in metal foam is complex and depends on the BRDF of skeleton surface strongly. The prediction of reflectivity and BRDF is an attractive issue in the radiative transfer calculation. In order to obtain the reflectivity and BRDF of skeleton
surface, the model used to simulate the real rough surface and suitable
method are crucial questions for the prediction of metal foam’s radiative properties. In this work, a method of modeling the skeleton rough
surfaces is introduced to further analysis of radiative transfer in metal
foam. This method is adapted to model and analyze the skeleton’s
surface of electrolytic Ni metal foam then. The FDTD method is adapted
to calculate the reflectivity and BRDF in the near infrared band. For
complete analysis of the radiative properties of skeleton’s rough surfaces, different size parameters of 1, 3 and 6 are compared and the
incident angles range from 0° to 60°. The conclusions can be summarized as followed.
1. The reflectivity of skeleton’s rough surfaces increases with decreasing size parameter and the reflectivity of different incident
angles are similar for a certain size parameter. For a larger incident
angle, the reflectivity decreases for all the size parameters. The reflectivity is affected by the size parameter obviously and is changed
slightly by different incident angles.
2. For the BRDF of skeleton’s surfaces, there are main peaks that own
most of reflected power near the mirror directions of incident lights.
Slight deviations of location appear when incident angles become
larger. With the size parameter increasing, the values of main peaks
increase obviously while the scopes of main peaks decrease. The
sub-peaks exit in different locations surrounding the main peaks fuse
gradually with increasing size parameter. When the size parameter
become 6, the sub-peaks vanish completely and become whole regions with isotropic reflection surrounding the main peaks.
3. For other directions in the hemispherical space, the reflections vary
from anisotropic reflection to isotropic reflection. For the largest
size parameter discussed here, the reflection in other directions can
be treated as the diffuse reflection.
For larger size parameters, the reflection of rough surface can be
simplified into the combination of mirror reflection and diffuse
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