Effect of Surface Roughness in Micro-nano Scale on Slotted Waveguide Arrays in Ku-band
Effect of Surface Roughness in Micro-nano Scale on Slotted Waveguide Arrays in Ku-band
Na LI 0 1 2
Peng LI 0 1 2
Liwei SONG 0 1 2
0 Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University , Xi'an 710071 , China
1 Supported by National Natural Science Foundation of China (Grant Nos. 51305322, 51405364, 51475348)
2 Effect of Surface Roughness in Micro-nano Scale on Slotted Waveguide Arrays in Ku-band
Modeling of the roughness in micro-nano scale and its influence have not been fully investigated, however the roughness will cause amplitude and phase errors of the radiating slot, and decrease the precision and efficiency of the SWA in Ku-band. Firstly, the roughness is simulated using the electromechanical coupled(EC) model. The relationship between roughness and the antenna's radiation properties is obtained. For verification, an antenna prototype is manufactured and tested, and the simulation method is introduced. According to the prototype, a contrasting experiment dealing with the flatness of the radiating plane is conducted to test the simulation method. The advantage of the EC model is validated by comparisons of the EC model and two classical roughness models (sine wave and fractal function), which shows that the EC model gives a more accurate description model for roughness, the maximum error is 13%. The existence of roughness strongly broadens the beamwidth and raises the side-lobe level of SWA, which is 1.2 times greater than the ideal antenna. In addition, effect of the EC model's evaluation indices is investigated, the most affected scale of the roughness is found, which is 1/10 of the working wavelength. The proposed research provides the instruction for antenna designing and manufacturing.
Slotted waveguide arrays; Roughness model; Micro/nano-scale; Amplitude and phase errors; Radiation characteristics
A Slotted Waveguide Array(SWA) antenna has the unique
advantages of having a compact configuration, stable
mechanical characteristics, low loss and high-efficiency, and
is consequently widely used in communication systems.
However, any structural deficiencies present, such as the
surface error on slots and planes [1, 2], have a direct
influence on its electrical properties. Although the surface
error can be reduced to its limit value by machining the
surface as flat as possible, roughness on a micro/nano-scale
is inevitable [3?5]. Since a major functionality requirement
is that the antenna is expected to operate at high
frequencies such as the Ku band, the amplitude of roughness is
equivalent to the working wavelength. In this case,
roughness will result in amplitude and phase errors of
radiating slots, and affect the self-admittance, coupling
relationship, and the matching condition of slots, which can
adversely affect the antenna?s electrical properties [6, 7].
Since this type of antennas is currently developed for
highfrequency bands, high gain, low side-lobe level, high
performance, ultra-wide band and high precision, the influence
of roughness on SWA is becoming a hot topic for research.
Because the electrical performance of a SWA is directly
affected by the degradation of its structural characteristics,
some researchers have explored different structural factors
that influence the antenna?s electrical properties. In terms
of general array antennas, the prime interest has been on
determinations of the pointing gain loss. On this subject,
RUZE  published the first work related to this field and
pursued issues regarding the effect of the position and
amplitude-phase errors of radiating elements on the
antenna gain loss. HSIAO  extrapolated that the effect
formula of error on the beam?s width, which provided a
very beneficial and applicable supplement to extant
theories on antenna gain loss. Subsequently, WANG 
investigated the influence of random errors for each
radiating element on the performance of a phased array antenna
based on the probability method. However, both RUZE and
WANG assumed that the structural error was within a
priori determined distribution and failed to analyze
practical structural deficiencies through a finite element
analysis of the antenna structure. Recently, TAKAHASHI,
et al. , and SONG, et al. , investigated the
dynamics related to distortions of the radiating surfaces and
their impact on the antenna?s electrical performance.
However, the majority of research concentrated on the
relation between radiating slot information and the
electrical performance of the antenna, and an extensive number
of research papers has been published on this topic.
Research has also been conducted on the relationship
between radiating slot information and cavity errors, but no
concrete associations have yet been determined. Moreover,
the roughness of the inner wall of the radiating waveguide
has hardly been studied.
Through his modeling research on the roughness of the
waveguide, MORGAN  obtained a result which is now
considered classical. The following analyses are found to be
consistent with MORGAN?s results [14?16]. However, in
MORGAN?s and the other analyses, some periodic functions
were used. In the other hand, TSANG, et al. , used a
random function to simulate roughness, which was
characterized using the root mean square(RMS), correlation length,
and the correlation function. Certainly, an advantage of
using a random model was that it allows a similar approach
as in the case of roughness occurring in copper
interconnects. LUKIC and FILIPOVIC  modeled a
rectangularcoaxial roughness by investigating the cubical,
semi-ellipsoidal and pyramidal indentation, and his results showed that
roughness accounted for up to 9.2% of their overall loss for
frequencies below 40 GHz. Nonetheless, the roughness on
the radiating waveguide is neither totally random nor clearly
deterministic, and the altitude distribution, the slope and
curve of the random model are associated with resolution
and sampling length of the measuring instrument, while it is
not unique [19?21]. Fractal geometry has provided an
additional means of description and roughness analysis
[22?24]. In our previous works, a one-dimensional fractal
analysis of roughness is investigated , but the scope of
its applicability is limited, because it is based on a
deterministic mathematical form.
Based on previous studies, this paper seeks to
investigate the relationship between surface roughness and
antenna electrical properties. For this reason, an EC model
was used to simulate roughness, based on which an
influence mechanism equation was deduced.
2 Roughness Model
2.1 Electromechanical Coupled Roughness Model
In the mechanical field, roughness is the surface profile of
an object separated from its surrounding environment,
which is the intersection transversal between the vertical
plane and actual surface, as shown in Fig. 1(a). However,
on the inner wall of a waveguide, the transmission path of
the electromagnetic wave is not completely consistent with
the surface profile. Even though roughness (black line in
Fig. 1(b)), may be bigger or smaller relative to the working
wavelength of the waveguide, the surface profile has little
influence on the transmission path (red line in Fig. 1(b))
. Therefore, an effective model should be built
according to the most influential surface profile, and profile
elements that are too big or too small should be considered
as sources producing noise and this noise should be filtered
out. Since roughness is a basic mechanical parameter,
while filtering concerns is the electrical properties, the
proposed novel roughness modeling method can be called
the electromechanical coupled roughness model and
abbreviated as the EC model.
In order to model roughness, a Monte Carlo simulation
was first used . Since two dimensional roughness
Fig. 1 Schematic diagram of the effective roughness model
exhibits waviness in both directions, if we denote the root
mean square (RMS) value of roughness as Ra, then the
power spectral density function of the fractal function is
S?wx; wy? ? hG2?D 1?.?2 ln?cxcy? ?wxwy??5 2D??i;
where D is the fractal dimension, and G is the characteristic
length, cx ? 1=Lx,cy ? 1 Ly, where Lx and Ly are the
sampling lengths of roughness in the x and y directions, while wx
and wy are the frequencies of the roughness in the x and y
directions, respectively. If the power spectral density
functions in each direction are S(wx) and S(wy), respectively, and
Ra / GD 1, then Eq. (1) can be expressed as
We performed a uniform discretization of roughness,
where the numbers of discrete points in each direction are
M and N, the corresponding separation distances of
adjacent points are Dx and Dy, where Dx ? Lx=M, Dy ? Ly N,
and the height of each data points on the surface is given by
1 XM=2 XN=2
exp?i?kmkxm ? knkyn??;
xm ? mDx; yn ? nDy, m ? 1 M=2 * M=2,
N=2 * N=2, while F?kmk; knk? is given by
where kmk ? 2pmk=Lx and knk ? 2pnk Ly.
For f ?xm; yn? to be real, F?kmk; knk? should satisfy the
where F ? ? is the Fourier transformation. Therefore, the
roughness model f ?x; y? can be expressed as
and the one dimension roughness model are
f ?x? ? GD 1 X1 cos?2pcnx ? un?;
f ?y? ? GD 1 X1 cos?2pcmy ? um?:
As shown in Fig. 2(a), a two-dimensional
WeierstrassMandelbrot (W-M) fractal function was used as the basic
model to describe the surface profile as comprehensively as
kmax ? 2pffi1ffiffi=ffiffi?ffiffipffiffiffifffiffilffiffiffi0ffirffiffiffi?ffi;
kmin ? kmax=3;
Fig. 2 Two dimensional roughness model
possible . Then, the effective roughness profile was
determined based on the transmission path, and the filter
function?s upper and lower limits were chosen. Finally, the
EC model was built as shown in Fig. 2(b). The specific
modeling parameters were detailed and illustrated in
subsequent equations. A two-dimensional Gaussian was used
as the filter function:
h?x; y? ? A2k1xky exph p?x=Akcx?2 p y Akcy 2i; ?8?
where kx and ky are the working wavelengths in the x and y
directions, while kcx, kcy and A are the truncation
wavelengths and width coefficient of the filter, respectively.
According to the electrical characteristics of the
antenna, 94% of the surface current was distributed within
3 times the skin depth, so the upper and lower limits of the
the f is the current frequency of the waveguide, and l0 and
r are the permeability and conductivity of the waveguide?s
inner wall, respectively.
The basic model given in Eq. (5) can be divided into
two parts as
f ?x; y? ? f1?x; y? ? f2?x; y?;
where f(x, y) is the basic model, which is composed of two
parts: f1(x, y) is the representationof noise in the surface
profile, which has little influence on the transmission path
of electromagnetic wave, and f2(x, y) is the effective profile
data, which is just the EC roughness model. Using f1(x, y)
as the assessment base level:
If the measured data are discrete, the assessment base
level fi1;j becomes
zi k;j lhk;lDxDy;
; LX M; j ? N; ; LY
p?kDx=akcx?2 p lDy akcy 2i.
The EC model fi2;j is
2.2 Evaluation indices of the EC roughness model
Traditional definitions of the surface roughness usually
contain only local and one-dimensional information, and
the arithmetical mean error Ra is regarded as the unique
evaluation indices . The features of two or three
dimensional roughness cannot be fully represented by this
conventional index. Moreover, Ra is a non-deterministic
index, which means that different roughness contours
processed by different technologies may yield the same Ra.
For the fractal geometric model, two parameters(D and G)
are introduced to describe roughness characteristics. Index
D is the dimension parameter, while G is the fundamental
frequency in space and represents roughness density.
Although these two indexes allow more accurate roughness
quantification, they cannot be adequately measured directly
and it is hard to relate them to the measured parameters of
roughness. For these reasons, the applicability of the fractal
model is limited. Studies have shown that the main
differences between roughness contours processed by
different technologiesbut with the same Ra are the density and
regularity of peaks and valleys. The information appears as
the expansion length of the roughness profile, which is the
principal reason for changing of the transmission path.
Therefore, two parameters, Ra and Rl are introduced as the
roughness model indexes. Ra is the main parameter, and its
value is equal to the arithmetical mean deviation; Rl is the
auxiliary parameter, and its value is the length of effective
roughness contour. The equations for these two indices are
8 1 Z lx 1 Xn
>>>>>>> Ra?1? ? lx 0 jf ?x?jdx n i?1 jyij ;
1 Z lx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where lx and f(x) are the sampling length and roughness
model in the x direction, while the (xi, yi) is a discrete
measured datumof roughness.
Using the two-dimensional roughness indices as the
description model can not only reflect the relief intensity
but also the fluctuationgradientfor roughness, which
required for both mechanical and electromagnetic field
analyses. Even though using a calculation based on a
onedimensional model displays a clear mathematical principle
and its simple structural equation is easily calculated, it is
revealed to be inadequate since the electromagnetic field
dispersed over the waveguide inevitably demonstrates
some directional properties. Furthermore, the real surface
roughness is anisotropic and should be considered
simultaneously from the length wise and transverse directions.
Thus, [Ra(2), Rl(2)] for the two-dimensional EC model are
formulated as follows:
1 Z lx Z ly
1 Z lx Z ly sffiffiffiffiffiffiffiffiffiffiffiffioffiffiffifffiffiffiffiffi2ffiffiffiffiffiffiffiffiffioffiffiffifffiffiffiffiffi2ffiffi
jf ?x; y?jdxdy;
3 Factors Affecting Roughness
In the field of antenna error analysis, there existed a
number of papers dealing with the effect of imperfections
in the waveguide cavity on radiating the slots? error. For an
antenna working in the GHz range, roughness of the
waveguide becomes a critical reason for position and
directional offset of the radiation slot. Therefore, in order
to establish a connection between the roughness and
radiating slot error, the key difficulty lies in antenna error
analysis. Based on the model f(x, y), the roughness
information should be primarily represented using antenna
Given a point on the roughness surface A(xA, yA, zA) and
an angle a, an elliptical curve can be obtained using a line
drawn through the point A, whose axis is the origin and a is
the angle. Let the center of this ellipse be Q(xQ, yQ, zQ) and
an intersection point of the axis and the ellipse be denoted
by S(xS, yS, zS). Thus, an area corresponding to the
roughness coordinate d?xdn; yd; zdn? is established, where the
point Q is its origin and the elliptical surface is on the xdnoydn
plane. If the antenna coordinate is (x, y, z), the
transformation process needs to be shifted twice and rotated as
x0 1 0 1 0 0 10 xnd 1
@ y0 A ? @ 0 cos xx sin xx A@ ynd A;
z0 ? zQ 0 sin xx cos xx zdn
0 x ? xQ 1 0 cos xz sin xz 0 10 x0 1
@ y ? yQ A ? @ sin xz cos xz 0 A@ y0 A;
z 0 0 1 z0
where xx ? p=2 x0x, and x0x can be deduced by
tan x0x ? cos sbi?nbcos a, while (x0, y0, z0) is the inter mediate
coordinate during the transformation process.
If the roughness model is zdn?xnd; ydn?, then according to
Eqs. (18), (19), the slot error characteristics are
xn2d ? yn2d ? z2nd ;
fhn ? arccos?z=x?
fun ? arctan?y=x?
If the excitation current of unit n is In, its coordinate are
(xn, yn, zn), and In is symmetrical on two coordinate axes,
the radiation field intensity of the antenna on the plane
(u ? u0) is 
X In exp j?k?xndx cos u0 ? yndy sin u0?
where E0?h; u0? is the far field pattern, h; u are the
azimuth in the far field, dx and dy are the distances of the array
element, xn and yn are the position coordinates of unit n,
and k is the transfer constant. When the position error of
X Jn exp j?k??xn ? xnd?dx cos u0?
4 Testing of Proposed Methods
4.1 Experiment on a planar slot antenna
In order to provide evidence supporting the EC roughness
model and how factors related to its influence antenna
performance, a miniature planar slot array antenna was
processed as an experimental project case. The antenna
operated in the Ku band, with a central frequency of
12 GHz, a gain of no less than 17 dB, and the first lobe
level was no higher than -16 dB. Its structural dimensions
were 150 mm 9 126 mm, and it had ten radiating
waveguides and eight vertical offset slots on each
Roughness on the inner wall of the radiating waveguides
was measured using the Taylor Hobson profile measuring
instrument. Indexes Ra and Rl of the measurement data
[Ra1, Rl1] and the EC roughness model [Ra2, Rl2] were
calculated as follows:
1 Z lx
In the equations, w2 is the measuring resolution, w1 is
the sampling length, D is the fractal dimension, and G is
the characteristic length. To validate the EC model, three
regions(S1, S2 and S3) on the radiating surface were
chosen, and the indexes of the measurement data and the
EC model on these regions were shown in Table 1. The
RMS value for S1 is 0.495 mm, for S2 it is 1.477 mm and
for S3 it is 0.703 mm, with a maximum error of 13%,
which demonstrates the accuracy of the EC model.
The antenna was fixed on the test turning platform to
ensure that the radiating plane was parallel to the direction
of gravity and the scanning plane of the measuring
waveguide probe. The near-field data of the antenna was
measured by the plane near-field scanning method, while
the far-field data were obtained through the near-field to
far-fieldtransformation(nffft) method [32, 33]. The plotting
radiating pattern is for the H-plane. The electrical
performance was evaluated using the antenna gain, the
maximumsidelobe level, and the 3-dB beamwidth on azimuth
plane, as well as the maximum sidelobe level, and 3-dB
beamwidth on the pitch plane.
4.2 Simulation of the SWA with Roughness
High-precision characteristics of SWA result in high costs
and a long manufacturing cycle. Thus, manufacturing
antennas in large quantities for experimental reasons is
difficult and even impractical. Therefore, antenna
simulation analysis is gaining increasing attention and interest
from the scientific and industrial community, because of
repeatability and low cost. However, the key to accurate
simulations is to incorporate surface roughness in the
numerical model. Only a small number of existing
commercial software can accurately analyze the antenna taking
roughness into account, and the structural and
electromagnetic analyses are usually carried out separately, which
will cause a mismatch in structural and electromagnetic
meshes. In order to deal with these issues, the grid
conversion and roughness data adding methods are presented
in this paper. As shown in Fig. 3, the simulation was
conducted as follows:
First, the finite element model of the antenna was
constructed using ANSYS 11.0, as shown in Fig. 4. The
number of nodes was 19263, while there were 55169 units.
The material chosen was Al-6063, with a modulus of
elasticity of 70 GPa, a Poisson ratio of 0.33, and a density
of 2.7 9 103 (kg m-3).
The second step was to add the roughness information to
the ideal antenna model. The structural boundary of the
antenna was determined by the shape of the data stream,
and the roughness model was added into it as an additional
Fig. 3 Simulation process of the antenna with roughness
Fig. 4 Finite element model of the antenna array
boundary. The finite element model of the antenna with
roughness was built using the GUI.
The third step was to translate the structural model to an
electromagnetic analysis model by converting the
tetrahedral body elements of the structural model into triangular
surface elements, which were necessary for the
electromagnetic analysis. Based on the surface elements, a surface
model of the antenna was built and the intracavity model
was extracted from it. Finally, the intracavity model of the
antenna was introduced into HFSS 11.0 and the electrical
properties of the model were obtained.
4.3 Results and Discussion
To verify the accuracy of the proposed simulation method,
a flatness comparison of the radiating plane was made
between the simulation results and the test data, as shown
in Fig. 5, where the series of radiating slots forms the
Fig. 5 Flatness comparison of the simulation result to the test data
abscissa and the z component was the vertical coordinate.
The simulation results along with the slot numbers are all
highly consistent with the test data results. A small error
occurs, because the simulation was based on an antenna
working under ideal circumstances, which was not the case
in the practical antenna test.
To verify the proposed roughness model, a comparison
was made using the test data of the antenna pattern, where
the sine wave represents the period model and the Gaussian
represents the random model. As shown in Fig. 6, the
results of the periodic model(based on the sine wave
approximation) have a lower first side-lobe than the test
data, because most of the roughness information has been
lost. The results obtained from the performance
degradation analysis of the fractal model appears more severe than
the result of the periodic models, due to the fact that the
fractal model contains more roughness information, which
severely changes the transmission path of the
electromagnetic wave. However, it does not completely coincide with
the test data, because it is built according to the actual
surface profile, not the transmission path. The results of the
EC model are closest to the test data, indicating that it can
describe the roughness information more precisely than
any of the other models can.
As shown in Fig. 7, increasing the roughness indices
also broadens the beam width and raises the side-lobe level
of the antenna, while it had little impact on the antenna?s
gain. The antenna is affected the most when the amplitude
of roughness is equal to approximately one tenth of the
working wavelength. Relative to Ra, Rl has a stronger
influence on the first side-lobe level of the antenna. These
findings suggest that the strongest influence indexes, and
reducing them in the manufacturing process is an effective
way to ensure the design accuracy of the antenna, which
will greatly reduce the production cost and shorten the
production cycle as well.
An EC roughness model is presented to characterize
the roughness by using the Gaussian filter. Two
evaluation indices of the EC model are introduced.
ARelationship between the roughness and the array?s
radiation properties is obtained by analyzing the
positional difference of the radiating slots.
A novel simulation method for the non-ideal antenna
Existence of roughness strongly broadens the beam
width and raises the side-lobe level of SWA, with the
maximum value being 1.2 times greater than
obtained by a smooth antenna.
The most affected scale of the roughness is found,
which is 1/10 of the working wavelength.
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1. CHEN X , HUANG K , XU X. A Novel planar slot array antenna with omnidirectional pattern [J]. IEEE Transactions on Antennas & Propagation , 2011 , 59 ( 12 ): 4853 - 4857 .
2. LIU F , XU G , LIANG L ,et al. Least squares evaluations for form and profile errors of ellipse using coordinate Data [J]. Chinese Journal of Mechanical Engineering , 2016 , 29 ( 5 ): 1 - 9 .
3. CHENG Z , LIAO R. Effect of surface topography on stress concentration factor [J]. Chinese Journal of Mechanical Engineering , 2015 , 28 ( 6 ): 1141 - 1148 .
4. BARBARINO S , FABRIZIO C. Effect of the substrate permittivity on the features of a UWB planar slot antenna [J]. Microwave & Optical Technology Letters, 2010 , 52 ( 4 ): 935 - 940 .
5. HESSAINIAA Z , BELBAHA A , YALLESEA M , et al. On the prediction of surface roughness in the hard turning based on cutting parameters and tool vibrations [J]. Measurement , 2013 , 46 ( 5 ): 1671 - 1681 .
6. MORINI A , ROZZI T , VENANZONI G. On the analysis of slotted waveguide arrays [J]. IEEE Transactions on Antennas & Propagation , 2006 , 54 ( 7 ): 2016 - 2021 .
7. MONTISCI G , MAZZARELLA G , CASULA G. A. Effective analysis of a waveguide longitudinal slot with cavity [J]. IEEE Transactions on Antennas & Propagation , 2012 , 60 ( 7 ): 3104 - 3110 .
8. RUZE J. Pattern degradation of space fed phased arrays [R]. M.I.T. Lincoln Laboratory , Lexington, MA, Project report SBR1 , 1979 .
9. HSIAO J. Array sidelobes, error tolerance, gain and beamwidth[R] . NRL Report 8841 , Interim Report Naval Research Lab., Washington, DC. Electromagnetics Branch , 1984 .
10. WANG H S C. Performance of phased-array antennas with mechanical errors [J]. IEEE Transactions on Aerospace & Electronic Systems , 1992 , 28 ( 2 ): 535 - 545 .
11. TAKAHASHI T , NAKAMOTO N , OHTSUKA M , et al. Onboard calibration methods for mechanical distortions of satellite phased array antennas [J]. IEEE Transactions on Antennas & Propagation , 2012 , 60 ( 60 ): 1362 - 1372 .
12. SONG L W , DUAN B Y , ZHENG F , et al. Performance of planar slotted waveguide arrays with surface distortion [J]. IEEE Transactions on Antennas & Propagation , 2011 , 59 ( 9 ): 3218 - 3223 .
13. MORGAN P. Effect of surface roughness on eddy current losses at microwave frequencies [J]. Journal of Applied Physics, 1949 , 20 ( 4 ): 352 - 362 .
14. HOLLOWAY C L , KUESTER E F. Power loss associated with conducting and superconducting rough interfaces [J]. IEEE Transactions on Microwave Theory & Techniques , 2000 , 48 ( 10 ): 1601 - 1610 .
15. GU X X , TSANG L , BRAUNISCH H. Modeling effects of random rough interface on power absorption between dielectric and conductive medium in 3-D problem [J]. IEEE Transactions on Microwave Theory & Techniques , 2007 , 55 ( 3 ): 511 - 517 .
16. WU Z , DAVIS L E. Surface roughness effect on surface impedance of superconductors [J]. Journal of Applied Physics, 1994 , 76 ( 6 ): 3669 - 3672 .
17. TSANG L , BRAUNISCH H , DING R. H, et al. Random rough surface effects on wave propagation in interconnects[J] . IEEE Transactions on Advanced Packaging , 2010 , 33 ( 4 ): 839 - 856 .
18. LUKIC M V , FILIPOVIC D S. Modeling of 3-D Surface roughness effects with application to l-coaxial lines [J]. IEEE Transactions on Microwave Theory and Techniques , 2007 , 55 ( 3 ): 518 - 525 .
19. CHEN Q , CHOI H W , WONG N. Robust simulation methodology for surface-roughness loss in interconnect and package modelings [J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits And Systems , 2009 , 28 ( 11 ): 1654 - 1665 .
20. GUO X C , JACKSON D R , KOLEDINTSEVA M Y , et al. An analysis of conductor surface roughness effects on signal propagation for stripline interconnects [J]. IEEE Transactions on Electromagnetic Compatibility , 2014 , 56 ( 56 ): 707 - 714 .
21. DING R H , TSANG L , BRAUNISCH H. Wave propagation in a randomly rough parallel-plate waveguide, microwave theory and techniques[J] . IEEE Transactions on Microwave Theory & Techniques , 2009 , 57 ( 5 ): 1216 - 1223 .
22. PERROTTI V , APRILE G , DEGIDI M , et al. Fractal analysis: a novel method to assess roughness organization of implant surface topography [J]. International Journal of Periodontics & Restorative Dentistry , 2011 , 31 ( 6 ): 633 - 639 .
23. MAJUMDAR A , BHUSHAN B. Role of fractal geometry in roughness characterization and contact mechanics of surfaces [J]. Journal of Tribology , 1990 , 112 ( 2 ): 205 - 216 .
24. ZHAO H ,WU Q. Application Study of fractal theory in mechanical transmission [J]. Chinese Journal of Mechanical Engineering , 2016 , 29 ( 5 ): 871 - 879 .
25. LI N , ZHENG F. Effect of micro/nano-scale rough surface on power dissipation of the waveguide: model and simulate[J] . Journal of Nanoscience & Nanotechnology , 2011 , 11 ( 12 ): 11222 - 11226 .
26. DING R , TSANG L , BRAUNISCH H. Random rough surface effects in waveguides using mode matching technique and the method of moments [J]. Components Packaging & Manufacturing Technology IEEE Transactions on , 2012 , 2 ( 1 ): 140 - 148 .
27. ZHANG J , GUO F. Statistical modification analysis of helical planetary gears based on response surface method and monte carlo simulation [J]. Chinese Journal of Mechanical Engineering , 2015 , 28 ( 6 ): 1194 - 1203 .
28. JAHN R , TRUCKENBRODT H. A simple fractal analysis method of the surface roughness [J]. Journal of Materials Processing Technology , 2004 , 145 ( 1 ): 40 - 45 .
29. WHITEHOUSE D J. Surfaces and their Measurement[M]. Oxford : Elsevier, Butterworth-heinemann, 2004 .
30. JAMNEJAD-DAILAMI V , SAMII Y R. Some important geometrical features of conic-section-generated offset reflector antenna [J]. IEEE Transactions on Antennas & Propagation , 1980 , 28 ( 6 ): 952 - 957 .
31. HUNG Y . Impedance of a narrow longitudinal shunt slot in a slotted waveguide array [J]. IEEE Transactions on Antennas & Propagation , 1974 , 22 ( 4 ): 589 - 592 .
32. OLINER A. The impedance properties of narrow radiating slots in the broad face of rectangular waveguide: Part I -theory[J] . IEEE Transactions on Antennas & Propagation , 1957 , 5 ( 1 ): 4 - 11 .
33. QURESHI M A , SCHMIDT C H , EIBERT T F. Efficient nearfield far-field transformation for non redundant sampling representation on arbitrary surfaces in near-field antenna measurements [J]. IEEE Transactions on Antennas & Propagation.
Na LI , born in 1982, is currently a associate professor at Key Laboratory of Electronic Equipment Structure Design , Ministry of Education, Xidian University, China. She received her PHD degree from Xidian University , China, in 2012 . Her research interests include surface error modeling, surface error reconstruction and the functionality design of the surface . Tel: ? 86 - 029 -88203040; E-mail: lina@mail .xidian.edu. cn Peng LI , born in 1981, is currently a associate professor at Key Laboratory of Electronic Equipment Structure Design , Ministry of Education, Xidian University, China. His research interests include multidisciplinary design optimization of the Slotted Waveguide Array . Tel: ? 86 - 029 -88203040; E-mail: yinhong0523 @163. com Liwei SONG , born in 1981, is currently a associate professor at Key Laboratory of Electronic Equipment Structure Design , Ministry of Education, Xidian University, China. His research interests include structural error analysis and error compensation of the Slotted Waveguide Array . Tel: ? 86 - 029 -88203040; E-mail: slw1206 @163.com