over 100 m MMF RoF with an EVM 4% and it is possible to

obtain 10�3 BER at �12 dBm received optical power.

ACKNOWLEDGMENTS

V. Sittakul would like to acknowledge Royal Thai Government

Scholarship for PhD funding.

REFERENCES

1. V. Sittakul and M.J. Cryan, A fully bidirectional 2.4 GHz wireless-

over fibre system using photonic active integrated antennas

(PhAIAs), IEEE J Lightwave Technol 25 (2007), 3358–3365.

2. V. Sittakul and M.J. Cryan, A 2.4 GHz wireless-over-fibre system

using photonic active integrated antennas (PhAIAs) and lossless

matching circuits, IEEE J Lightwave Technol, in press.

3. V. Sittakul and M.J. Cryan, A comparison of single mode and mul-

timode fibre links for use in wireless-over-fibre systems, Micro-

wave Opt Technol Lett 51 (2009).

4. M. Boylmalf, A. Sobh, and S. Akhtar, Physical layer performance

of 802.11 g WLAN, In: Proceedings of the Applied Telecommuni-

cations Symposium, Washington, DC, 2004, pp. 175–178.

5. C. Heegard, Range versus rate in IEEE 802.11g wireless local area

networks, In: Proceedings of the IEEE 802.11 Task Group G, Seat-

tle Washington, September 2001.

6. M. Mjeku and N.J. Gomes, Analysis of the request to send/clear to

send exchange in WLAN over fiber networks. J Lightwave Technol

26 (2008), 2531–2539.

7. Available at: http://www.ieee802.org.

8. G. Singh and A. Alphones, OFDM modulation study for a radio-

over-fiber system for wireless LAN (IEEE 802.11a), In: Proceedings

of the Communications and Signal Processing and the Pacific Rim

Conference on Multimedia, vol. 3, Singapore, 2003, pp. 1460–1464.

9. F. Tabatabai and H.A. Raweshidy, Performance evaluation for

WLAN and UWB using radio over fibre, In: Proceedings of the

9th European Conference on Wireless Technology (ECWT 006),Manchester, UK, 2007, pp. 147–149.

10. S.D. Personick, Baseband linearity and equalization in fiber optic

digital communication systems, Bell Syst Tech J 52 (1973),

1175–1195.

11. C.H. Cox, Analog optical link theory and practice, Cambridge Uni-

versity Press, Cambridge, England, 2004; Chapter 5, pp. 169–178.

12. C.K. Sim, M.L. Yee, B. Luo, L.C. Ong, and M.Y.W. Chia, Per-

formance evaluation for wireless LAN, Ethernet and UWB co-exis-

tence on hybrid radio-over-fiber picocells, In: Proceedings of the

OFC/NFOEC JWA60 Conference, 2005.

13. S. Hwang et al., RoF technologies for in-building wireless systems,

IEICE Trans Electron E90-C (2007), 345–350.

VC 2010 Wiley Periodicals, Inc.

IMPROVED NURBS MOM-PO METHODFOR ANALYZING ANTENNA AROUNDELECTRICALLY LARGE PLATFORM

Kai Huang, Zhi-Li He, and Chang-Hong LiangSchool of Electronic Engineering, Xidian University, Xi’an, Shaanxi710071, People’s Republic of China; Corresponding author:huangkai841025@126.com

Received 2 March 2010

ABSTRACT: The hybrid method of moments and physical optics(MOM-PO) based on nonuniform rational B-spline (NURBS) modeling

represented a suitable approach for dealing with electromagnetic (EM)radiation problems because of its higher efficiency and more accurate

modeling than Triangle MOM-PO method. However, the invalidation inexisting NURBS MOM-PO method occurs when the distance betweenantenna and scatterer is less than one wavelength. The reason is

analyzed in detail in this article. This article introduces an improvedNURBS MOM-PO method to overcome this problem. Ludwig integralcombining with stationary phase method is utilized to calculate the

integral of PO currents. Thus, both the efficiency and validity areensured. Numerical results demonstrate that accurate results can be

achieved efficiently in electrically large EM radiation problems by usingthe present method. VC 2010 Wiley Periodicals, Inc. Microwave Opt

Technol Lett 52:2675–2679, 2010; View this article online at

wileyonlinelibrary.com. DOI 10.1002/mop.25569

Key words: nonuniform rational B-spline (NURBS); hybrid method ofmoments and physical optics (MOM-PO); Ludwig integral; electrically

large

Figure 8 Constellation diagram; without RoF (left), after 100 m MMF (right)

Figure 9 EVM versus optical received power

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 52, No. 12, December 2010 2675

1. INTRODUCTION

Triangle method of moments and physical optics (MOM-PO)

[1, 2] is a kind of widely adopted numerical method in dealing

with electrically large electromagnetic (EM) problems. This

approach usually uses a group of triangle facets to model a struc-

ture. More triangle facets are needed when the frequency

increases. As the memory requirement and CPU time increases

when the number of triangle facet increases, it could become

unaffordable when the frequency becomes relatively high.

Another drawback of this approach is that the triangle facet is not

good at representing a realistic object accurately. In 1992, Perez

and Catedra [3] first introduced the nonuniform rational B-spline

(NURBS) surface modeling technology into the field of high fre-

quency EM computation. This technology has many advantages

such as high in modeling precision, few in the number of surfa-

ces, and so on. Therefore, EM computational methods based on

NURBS modeling [4–7] have been widely analyzed since 1992.

In 2007, MOM-PO was utilized to calculate the disturbed radia-

tion pattern of antenna around a platform modeled by NURBS

surfaces by Chen et al. [7] and satisfying effects were obtained.

The advantage of NURBS MOM-PO method is that the execution

time is not proportional to the platform’s electrical size because

the number of the computation elements is not proportional to the

platform’s electrical size. It means the method is superior in effi-

ciency than conventional MOM-PO that based on triangle facet

model when the frequency is relatively high.

In the last part of Ref. 7, however, it is pointed out that

‘‘One important issue is how to deal with cases when antenna is

very close (less than one wavelength) or even connected to the

platform,’’ which implies that the method is invalid when the

distance between antenna and scatterer is less than one wave-

length. By careful analysis, we have found that the invalidation

is resulted from the stationary phase method (SPM) used in Ref.

7. In fact, SPM is invalid in the computation of near field, and

this invalidation leads to the incorrect impedance matrix ele-

ment, and then to the wrong current distribution over the

antenna, which is obtained by modifying the impedance matrix.

To overcome this invalidation, SPM might be replaced by Lud-

wig integral in calculating the PO currents in the case when

antenna is very close to the scatterer. In Ludwig integral, how-

ever, the integral region should be divided into small grids,

which is not as efficient as SPM because SPM is analytical.

Considering that the electrical size of the PO region is much

larger than that of the MOM region and not all of the patches

are less than 1.0 wavelength away from the antenna, a technique

that using Ludwig integral combined with SPM might be intro-

duced to ensure the efficiency and overcome the invalidation.

Concretely, a pretreatment is to be executed to test the distance

between the antenna and each Bezel surface. For a certain sur-

face, if the distance is less than 1.5 wavelengths, Ludwig inte-

gral is used and SPM is used otherwise. Numerical results show

that this method is efficient and accurate without restrictions on

the distance between the antenna and the surface. Thus, it could

be utilized to deal with the disturbed radiation pattern of

antenna near electrically large scatterer.

2. THEORY

2.1. MOM-PO Process Based on NURBS ModelingWhen the MOM-PO approach is applied to analyze the antenna

around a complex structure, the antenna is taken as the MOM

region and the structure as the PO region. In this article, the PO

region is modeled with NURBS surfaces as shown in Figure 1.

Considering the convergence issue, we use modifying impend-

ence matrix to implement the interaction of MOM region and

PO region. This hybrid technique is based on the electric field

integral equation (EFIF). Supposing the current over the MOM

region is ~JMold

and the one is ~JPO

over the PO region. The EFIF

is established in the MOM region:

L1ð~JMoMÞtan þ L2ð~JPOÞtan ¼ �~Einc

tan (1)

where L1 and L2 are the corresponding operators which trans-

form the currents into respective electric scatter fields. ~Einc

tan is

exciting source of the antenna. The unknown current coefficients

in the MOM region are approximated by a linear superposition

of basis functions:

~JMoM ¼ ~g½ � IMoM½ � ¼ ~g1 � � �~gn½ �

I1...

In

264

375 (2)

in which ~g1…~gn are the basis functions, I1 … In are the corre-

sponding coefficients. The induced current in the PO region can

be represented as:

~JPO ¼ LMoMð~JMoMÞ (3)

Through the inner product process, (3) is transformed to:

ZMoM½ � þ ZMoM�PO½ �ð Þ IMoM½ � ¼ V½ � (4)

where [ZMoM] is the self-impendence matrix of the MOM

region, and [ZMoM-PO] is the mutual impendence matrix between

MOM and PO regions. The voltage matrix is:

Vn ¼ wn;�~Einc

tan

D E(5)

where wn is the weight function. The impendence matrix can be

calculated as follows:

Zmn ¼ wm;L1ð~gnÞtan� �þ wm;LMoM�POð~gnÞtan

� �(6)

in which LMoM�POð~gnÞ ¼ L2LMoMð~gnÞ means the PO scattered

field with the nth MOM subdomain as source. Once [IMoM] is

solved, the property of the antenna can also be analyzed from

the current distribution in the MOM region.

Figure 1 MOM-PO process based on NURBS modeling. [Color

figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com]

2676 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 52, No. 12, December 2010 DOI 10.1002/mop

2.2. Analysis of the Invalidation ReasonAlthough the NURBS surface has advantages of modeling, it is

very difficult to be applied in the EM calculation because of the

utilization of basis functions of recursive form. Therefore,

NURBS surfaces used for EM calculation usually need to be

converted into Bezier surfaces. The basic expression for Bezier

surface is

~Sðu; vÞ ¼

Pni¼0

Pmj¼0

wij~PijB

ni uð ÞBm

j vð ÞPni¼0

Pmj¼0

wijBni uð ÞBm

j vð Þ; u; vð Þ 2 0; 1½ �2 (7)

where ~Pij stands for the control points of the surface, wij the

weights of the control points. The key point of the theory in

Ref. 7 is the calculation of the physical optical scattered fields

taking the nth MOM subdomain as the source, as shown in Fig-

ure 2. Supposing the MOM region is divided into N subdomains,

the incident magnetic field produced by the mth (m ¼ 1, 2, …N)MOM subdomain is:

~Hmð~rsÞ ¼ZSm

�ð1þ jkRsdÞð~Rsd �~gmÞ4pR3

sd

e�jkRsd dSn (8)

where Sm is definition domain of the mth basis function of

MOM region, k ¼ 2 pf/C, f is the antenna’s working frequency,

C is the velocity of light, ~Rsd ¼~rs �~rd, ~rd is the point within

the mth MOM subdomain, and ~rs is the point on the rational

Bezier surface. The induced current on the surface is obtained

from physical optics approximation:

~JPO ¼ 2n� ~H

i;

0;

in the lit PO region

in the dark PO region

�(9)

in which n is the unit normal vector of the surface. Through

several simple algebraic steps, the total PO scattered field from

NURBS surfaces is expressed as:

~EPO ¼ �d

16jxep2

Z 1

0

Z 1

0

~gðu; vÞejkf ðu;vÞdudv (10)

where

f ðu; vÞ ¼ � Rsd þ Rfsð Þ (11)

~gðu; vÞ ¼ 3� k2R2fs þ j3kRfs

R5fs

~Rfs��

~Rfs � 2n� ~Rsd �~gn� �� �

þ 4 1þ jkRfsð ÞR3fs

n� ~Rsd �~gn� �� �� ð1þ jkRsdÞ

R3sd

~rsu �~rsvj j (12)

in which ~rf is the observation point, ~Rfs ¼~rf �~rs, j~Rfsj is the

distance from the surface to the observation point, ~rsu ¼ @~rs@u and

~rsv ¼ @~rs@v . Thus, the core of the problem becomes to calculate a

surface integral such as

I ¼Z 1

0

Z 1

0

g u; vð Þejkf u;vð Þdudv (13)

It was presented that this integral can be computed by SPM

when k � 1 in Ref. 7. Actually, the condition under which

SPM is applicable is that the exponential part of the integrand

should be much more than 1. When the action of the scattered

field on the MOM element is implemented, Rsd, the distance of

the antenna to the scatterer, is almost equal to Rfs. Therefore,

when Rsd is small, the exponential part of the integrand in (13)

k f ðu; vÞj j ¼ k � Rsd þ Rfsð Þj j ¼ 2pk

� Rsd þ Rfsð Þj j (14)

cannot satisfy the condition. It directly results in the incorrect

impedance matrix element and then the wrong current distribu-

tion. On the basis of the above analysis and with a hope of

improving the invalidation, we attempt to calculate the scattered

field in near field region by using Ludwig integral instead of

the SPM.

2.3. Application of Ludwig Integral Combining with SPMLudwig integral [8, 9] is a very practical engineering numerical

integration method. To calculate Eq. (13) by Ludwig integral,

we expand g(u,v) and f(u,v) at four vertices of the integral do-

main by first-order Taylor’s series expansion. The expansion of

phase function f(u,v) is expressed as follows:

f ðu; vÞ � f ð0; 0Þ þ fu 0; 0ð Þuþ fv 0; 0ð Þv (15)

f ðu; vÞ � f ð1; 1Þ þ fu 1; 1ð Þ u� 1ð Þ þ fv 1; 1ð Þ v� 1ð Þ (16)

f ðu; vÞ � f ð0; 1Þ þ fu 0; 1ð Þuþ fv 0; 1ð Þ v� 1ð Þ (17)

f ðu; vÞ � f ð1; 0Þ þ fu 1; 0ð Þ u� 1ð Þ þ fv 1; 0ð Þv (18)

If we let all the partial derivatives to be approximated by differ-

ences and take an average of these four expressions, then f(u,v)can be written as

f ðu; vÞ ¼ aþ buþ cv (19)

where

a ¼ 1

43f 0; 0ð Þ � f 1; 1ð Þ þ f 1; 0ð Þ þ f 0; 1ð Þ½ � (20)

b ¼ 1

2f 1; 0ð Þ � f 0; 0ð Þ þ f 1; 1ð Þ � f 0; 1ð Þ½ � (21)

c ¼ 1

2f 0; 1ð Þ � f 0; 0ð Þ þ f 1; 1ð Þ � f 1; 0ð Þ½ � (22)

Similarly, the magnitude function g(u,v) can be approximated as

gðu; vÞ ¼ lþ muþ nv (23)

Figure 2 NURBS-PO scattered field when illuminated by subdomain

of MOM. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 52, No. 12, December 2010 2677

And the coefficients can be obtained as 20–22. Then the integral

(13) can be written as

I ¼Z 1

0

Z 1

0

lþ muþ nvð Þejk aþbuþcvð Þdudv (24)

which can be separated into the following three integrals:

I ¼ I1 þ I2 þ I3 (25)

where

I1 ¼Z 1

0

Z 1

0

l � ejk aþbuþcvð Þdudv ¼ l � ejka � ejkb � 1

jkbþ ejkc � 1

jkc

8>>:9>>;(26)

I2 ¼Z 1

0

Z 1

0

mu � ejk aþbuþcvð Þdudv

¼ m � ejka � ejkc � 1

jkc� ejkb

jkb� ejkb � 1

jkbð Þ28>>>:

9>>>; (27)

I3 ¼Z 1

0

Z 1

0

nv � ejk aþbuþcvð Þdudv

¼ n � ejka � ejkb � 1

jkb� ejkc

jkc� ejkc � 1

jkcð Þ28>>>:

9>>>; (28)

Substituting Eqs. (26–28) into (25), then the result of (13) can

be calculated finally. The integral domain is a rectangle.

Actually when using Ludwig integral, we usually divide the rec-

tangle into N � N small rectangles. In that case, we should deal

with the following expression

I ¼Z /mþ1

/m

Z hmþ1

hm

g u; vð Þejkf u;vð Þdudv (29)

At this time, the integral domain is one of the N � N small rec-

tangles. (ym, fm), (ym, fmþ1), (ymþ1, fmþ1), and (ymþ1, fm) are

the coordinates of the small square’s four vertexes, respectively.

SPM is more efficient than Ludwig integral because it is analyti-

cal and it does not need to divide the integral region. The basic

idea of SPM is that the contribution to the integral is mainly

from the vicinity of some certain critical points. The detail of

utilizing SPM can be seen in Ref. 7 clearly. To enhance the effi-

ciency of the present method, a technique that using Ludwig in-

tegral combining with SPM is introduced. A pretreatment is

executed to test the distance between the antenna and each

Bezel surface. For a certain surface, if the distance is less than

1.5 wavelengths, Ludwig integral is used and the SPM is used

otherwise. Thus, the present method can be used to analyze dis-

turbed pattern of antenna around electrically large platform

accurately and efficiently.

Figure 3 Radiation pattern of u ¼ 45� plane with antenna’s position

(0.21213, 0.21213, 0.0)

Figure 4 Radiation pattern of u ¼ 45� plane with antenna’s position

(0.17678, 0.17678, 0.0)

Figure 5 The head section of an airplane modeled with NURBS surfa-

ces and its control points. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com]

Figure 6 Radiation pattern of u ¼ 180� plane in Example 2

2678 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 52, No. 12, December 2010 DOI 10.1002/mop

3. NUMERICAL RESULTS AND DISCUSSION

The model in Ref. 7 is calculated by the present method at first.

The model of the spherical surface is in the inset of Figure 3, the

radius of which is 0.2 m. The antenna whose length is 0.5k and

working frequency 3 GHz is centered on (0.21213, 0.21213, 0.0)

and placed 1k away from the surface. The result of the disturbed

radiation pattern in the plane of f ¼ 45� under spherical coordi-

nates is compared with that in Ref. 7 as shown in Figure 3, which

shows good agreement with each other. This demonstrates that

the present method is applicable when the SPM is valid.

The same model is further analyzed when the center of the

antenna is moved to (0.17678, 0.17678, 0.0), hence, the distance

of the antenna to the surface is 0.5k. The calculated result of the

disturbed radiation pattern in the plane of f ¼ 45� is compared

with that obtained from NURBS MOM-PO SPM method in Ref.

7 and that from pure MOM, as shown in Figure 4. It is obvious

to see that the result of the present method agrees very well

with that of MOM, whereas SPM is invalid indeed. Therefore,

the present approach is very useful for obtaining accurate results

without restrictions on the distance between the antenna and the

surface.

In the last example, the present method is utilized to calcu-

late the pattern when the antenna is near the head section of an

airplane modeled with NURBS surfaces, which is shown in Fig-

ure 5. The control points and electrical sizes are also shown in

the picture. The length of the antenna is 2.0k, placed at (3.3,

0.0, 4.0) and operated in a frequency of 3.0 GHz. The number

of unknowns for the antenna is set as 40. This model contains

only four Bezier surfaces. The normalized radiation pattern of f¼ 180� plane is calculated and shown in Figure 6. The result is

in excellent agreement with that from Triangle MOM-PO.

Meanwhile, the number of the computation elements is not pro-

portional to the platform’s electrical size. So the present method

is still superior in efficiency than Triangle MOM-PO when the

frequency is relatively high.

4. CONCLUSIONS

This article has proposed an improved MOM-PO hybrid method

based on NURBS modeling for metallic structures. The reason

of the invalidation in the existing method, which occurs when

the distance between antenna and platform is less than one

wavelength, was analyzed in detail. This method uses Ludwig

integral combined with SPM in calculating the integral of PO

currents. The method avoids the limitation of SPM, while holds

its efficiency. The comparison of the results obtained by this

method and the ones in literatures and MOM shows that the

present method is practical in achieving accurate results without

restrictions on the distance between antenna and surfaces.

Besides, to obtain a more precise result, the multilevel physical

optics [10, 11] effect should be considered and the Fork current

[12] should also be adopted to modify the induced currents in

shadowed region. The authors will take this approach as their

future work. It will make sense to do further research about the

numerical computation based on NURBS modeling.

REFERENCES

1. U. Jakobus and F.M. Landstorfer, Improvement of the PO-MM

hybrid method by accounting for effects of perfectly conducting

wedges, IEEE Trans Antennas Propag AP-43 (1995), 1123–1129.

2. R.E. Hodges and Y. Rahmat-Samii, An iterative current-based

hybrid method for complex structures, IEEE Trans Antennas

Propag AP-43 (1997), 265–276.

3. J. Perez and M.F. Catedra, RCS of electrically large targets mod-

eled with NURBS surfaces, Electron Lett 28 (1992), 1119–1121.

4. J. Perez and M.F. Catedra, Application of physical optics to the

RCS computation of bodies modeled with NURBS surfaces, IEEE

Trans Antennas Propag 42 (1994), 1404–1411.

5. O.M. Conde, J. Perez, and M.F. Catedra, Stationary phase method

application for the analysis of radiation of complex 3D conducting

structures, IEEE Trans Antennas Propag 49 (2001), 724–730.

6. J.-L. Hu, S.-M. Lin, and W.-B. Wang, Computation of PO integral

on NURBS surface and its application to RCS calculation, Electron

Lett 33 (1997), 239–240.

7. M. Chen, Y. Zhang, X.W. Zhao, and C.H. Liang, Analysis of

antenna around NURBS surface with hybrid MOM-PO technique,

IEEE Trans Antennas Propag 55 (2007), 407–413.

8. A.C. Ludwig, Computation of radiation patterns involving numeri-

cal double integration, IEEE Trans Antennas Propag 11.11 (1968),

767–769.

9. W. Meng, W. Nan, and L. Chang-Hong, Problem of the singularity

of Ludwig’s algorithm, Microwave Opt Technol Lett 49 (2007),

400–403.

10. A. Boag, A fast physical optics (FPO) algorithm for high frequency

scattering, IEEE Trans Antennas Propag 52 (2004), 197–204.

11. L. Gurel and A. Manyas, Multilevel physical optics algorithm for

fast solution of scattering problems involving nonuniform triangu-

lations, In: Proceedings of the 2007 IEEE International Symposium

on Antennas and Propagation, Honolulu, Hawai’i, USA, June 2007.

12. U. Jakobus and F.M. Landstorfer, Application of FOCK-CUR-

RENTS for curved surfaces within the framework of a current-

based hybrid method, In: Computation in electromagnetics, April

10–12, 1996, pp. 415–420.

VC 2010 Wiley Periodicals, Inc.

EFFECT OF GROUND PLANE SIZE ONRADIATION PATTERN IN IFF/SSRMICROSTRIP ANTENNA ON THICKSUBSTRATE FED BY H TYPE SLOT

W. Zieniutycz,1 M. Mazur,2 and M. Pergol11 Faculty of Electronics, Telecommunications, and Informatics,Gdansk University of Technology, Narutowicza 11/12, 80-952Gdansk, Poland; Corresponding author: mper@ eti.pg.gda.pl2 Telecommunications Research Institute (PIT S.A.), Hallera 233A,80-502 Gdansk, Poland

Received 2 March 2010

ABSTRACT: Numerical and experimental study of the effect of the

ground plane size on radiation pattern of microstrip antenna on thicksubstrate is presented in the article. First the results of simulations andmeasurements of reference microstrip antenna are shown. Next the

ground plane sizes were reduced in both x and y directions and the sidelobes and backward lobe levels were calculated and measured. It was

shown how far we can reduce the size of ground plane withoutimportant deterioration of the antenna pattern. VC 2010 Wiley

Periodicals, Inc. Microwave Opt Technol Lett 52:2679–2682, 2010;

View this article online at wileyonlinelibrary.com. DOI 10.1002/

mop.25568

Key words: microstrip antenna; planar array; radiation pattern

1. INTRODUCTION

IFF systems (Identify Friend or Foe utilized by army) and SSR

systems (Secondary Surveillance Radar–used to support civil

air-traffic control) are widely used to identify flying objects [1].

Both civil and military ones are equipped with transponders.

Secondary IFF/SSR radars are sending impulses to them at fre-

quency 1.03 GHz. Their configuration determines type of in-

quiry. The transponders receive the information and generate

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 52, No. 12, December 2010 2679