frequency is above 11 KHz, the curve of h(f) is declined

abruptly. Figure 5(b) shows the amplitude-frequency response of

the FFP tunable filter. The amplitude-frequency of the filter is

kept under �2dB below 1.8 KHz, and the resonant frequency of

the filter is about 13 KHz for the minimum amplitude-frequency

response. For applications of this device to fast and accurate

scanning or locking, it is better to select the frequency of sinu-

soidal signal below 1.8 KHz.

4. CONCLUSIONS

In summary, a simple and an accurate method for measuring the

phase-frequency response of FFP tunable filters has been pro-

posed, which is based on a modification to the dynamic transfer

function of the filter. The method has been demonstrated suc-

cessfully with measurements of the phase-frequency response

and the amplitude- frequency response of a FFP tunable filter.

The phase-frequency characteristic is important for designs of

the device and surrounding control electronics, especially for the

wavelength locking application.

ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science

Foundation of China under Grant No. 60677024 and the National

High Technology Research Development Program of China under

Grant No. 2009AA03Z418.

REFERENCES

1. N.L. Taranenko, S.C. Tenbrink, K. Hsu, and C.M Miller, Fiber

Fabry-Perot tunable filter for high-speed optical packet switching,

Proc SPIE 2918 (1997), 26–37.

2. L.L. Wang, and T. Kowalcyzk, A novel locking technique for very

narrow tunable optical filters with sub-GHz 3-dB bandpass,

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3. Y. Ji, S. Zheng, Z. Li, X. Jin, X. Zhang, and H. Chi, Tunable fiber

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4. A.D. Kersey, T.A. Berkoff, and W.W. Morey, Multiplexed fiber

Bragg grating strain-sensor system with a fiber Fabry-Perot wave-

length filter, Opt Lett 18 (1993), 1370–1372.

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6. J.S. Patel, J.R. Wullert II, and M.W. Maeda, Frequency tracking of

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a high-finesse Fabry-Perot interferometer using the frequency

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11. Y. Yeh, D. Park, and S.H. Park, High-speed measurement of free

spectral range voltage of tunable filters, Opt Lett 34 (2009), 52–54.

12. M. Nakazawa, Phase-sensitive detection on Lorentzian line shape

and its application to frequency stabilization of lasers, J Appl Phys

59 (1986), 2297–2305.

VC 2011 Wiley Periodicals, Inc.

QUASIHYBRID SCHEME OFNONUNIFORM RATIONAL B-SPLINE–UNIFORM GEOMETRICAL THEORY OFDIFFRACTION METHOD FOR RADIATIONOF ANTENNAS

Nan Wang, Xiaojie Dang, Haobo Yuan, and Changhong Liang

Science and Technology on Antenna and Microwave Laboratory,Xidian University, Xi’an, Shaanxi 710071, China; Correspondingauthor: wangnan@mail.xidian.edu.cn

Received 1 March 2011

ABSTRACT: The nonuniform rational B-spline–uniform geometrical

theory of diffraction (NURBS-UTD) method is studied in this paper. Todeal with arbitrary wire antennas without analytical directional

functions, a quasihybrid scheme is presented to discuss the radiationpattern of wire antennas mounted around electrically large targets. TheNURBS technique is introduced to construct targets, the wire antennas,

and electrically large targets are analyzed by method of moment (MoM)and NURBS-UTD, respectively. Information from results of ray tracing

is chosen as interface between MoM and NURBS-UTD to formquasihybrid scheme. The practicability and validity can be seen fromexamples given. VC 2011 Wiley Periodicals, Inc. Microwave Opt Technol

Lett 53:2963–2967, 2011; View this article online at

wileyonlinelibrary.com. DOI 10.1002/mop.26374

Key words: NURBS; UTD; MoM; radiation pattern

1. INTRODUCTION

With the development of computer-aided geometric design, the

nonuniform rational B-spline (NURBS) technique becomes

another reliable way to construct targets for electromagnetic

simulation beside flat patches and analytical patches. As an uni-

versal computer-aided design technique, NURBS has the advan-

tages of high precision in modeling as illustrated in Figure 1

and great flexibility in forming algorithms.

In the area of electromagnetic simulation, the NURBS tech-

nique was first introduced into physical optics (PO) to calculate

radar cross section of electrically large targets [1, 2] and since

then, the NURBS-PO method was carefully studied in various

applications [3–5]. In the method of moment (MoM), lots of

studies have been made from basic thoughts to different basis

functions to try to absorb the NURBS technique in Refs. 6

and 7.

Uniform geometrical theory of diffraction (UTD) method is a

highly efficient method dealing with electrically large problems.

Theoretically, UTD can solve problems with arbitrary electrical

size without any extra need for computation resources. Due to

the successful introduction of NURBS technique into PO and

MoM, many works have been done on UTD method based on

NURBS (NURBS-UTD) to take advantage of the convenience

of NURBS in modeling [8–10]. In NURBS-UTD method, the

source can be point source, line source, and plane waves due to

their simplicity. Also, the source can be dipole or monopole,

because they have recognized analytical expression of direc-

tional functions. But, if the source is an antenna without analyti-

cal expression of directional function, the radiation of the

antenna will be hard to analyze only using NURBS-UTD.

To analyze complex wire antenna mounted around electri-

cally large targets while inheriting the advantages of NURBS-

UTD in dealing with electrically large problems, a quasihybrid

scheme is presented in this paper where MoM is used to analyze

wire antenna, and NURBS-UTD is used to analyze electrically

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 2963

large targets. The information (like information of reflected point

or diffracted point and so on) resulting from ray tracing is chosen

as interface between MoM and NURBS-UTD to form the quasihy-

brid scheme. Several examples are given in this paper to show the

validity of the quasihybrid scheme and the root-mean-square

(RMS) error is used to evaluate the results and show the precision.

2. QUASIHYBRID SCHEME

The mathematical expression of NURBS surface [11], which is

a kind of parametric surface, can be written as follows.

rðu; vÞ ¼Pn

i¼0

Pmj¼0 aijPijN

ipðuÞNj

qðvÞPni¼0

Pmj¼0 aijN

ipðuÞNj

qðvÞ���! u 2 ½0; 1�

v 2 ½0; 1��

(1)

where Pij is the control point, and aij is the corresponding weight

value, Nip(u) is the normalized B-spline basis function of degree p.

Any point on the surface can be derived from this expression. To-

gether with corresponding algorithm and the concept of differential

geometry, the geometrical information of any point on the surface

can also be derived easily, which becomes the joint between UTD

method and NURBS surfaces where basic aspects like ray tracing,

shadowing test, and calculating ray field are managed through nu-

merical measures, respectively [12, 13]. The NURBS-UTD method

which is based on the expression above can be applied directly to

arbitrary electrically large convex surfaces with highly accurate

approximation and can improve the precision of scattered field

especially when the surface has changed curvatures.

In NURBS-UTD method, all types of ray fields can be con-

cluded into the format given below

EðR0Þ ¼ EðQÞ � D¼ �AðsÞe�jks (2)

where R0 is the observation point; Q denotes the interactive

point, such as reflected point and diffracted point; D¼

is the

dyadic coefficient corresponding to different actions, such as the

dyadic reflected coefficient and the dyadic diffracted coefficient;

A(s) is the amplitude fading factor; and e�jks is the phase fading

factor. As can be seen in Figure 2, the first item, E(Q), in (2) is

the field traveling from the source point to the interactive point.

If a point source, line source is placed at the source point,

E(Q) is easily written as

EðQÞ ¼C e�jkr

r spherical wave

C e�jkrffiffir

p cylindrical wave

Ce�jkr plane wave

8><>: (3)

where C is a vector related to the strength of source.

Figure 1 NURBS surfaces. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]

Figure 2 Illustration of ray field

Figure 3 Major procedure of the quasihybrid method

Figure 4 An arbitrary surface with 16 control points

2964 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 DOI 10.1002/mop

If what is placed at the source point is simple wire antenna

like dipole or monopole which can be seen as a point source

without length, E(Q) is also easy to get by their recognized ana-

lytical directional function

EðQÞ ¼ Ccosðkl cos hÞ � coskl

sin h(4)

where C is a vector related to the strength of source, and l isthe half length of the dipole.

Problems appeared when the antenna placed at the source point

is complex, under this condition, the antenna may not have a ana-

lytical directional function, so that the radiation of antenna is really

hard to achieve only from NURBS-UTD. Aiming at arbitrary wire

antennas, a quasihybrid scheme is presented to combine the MoM

which is highly effective in analyzing wire antennas with NURBS-

UTD method. The main process of NURBS-UTD can be subdi-

vided into three parts, namely, ray tracing, shadowing test, and cal-

culating ray field. Given a source point and a observation point, the

ray-tracing process of NURBS-UTD is first taken, and all possible

kinds of rays are traced to find interactive point Q and the dyadic

coefficient D¼

and other useful information; second, the ray path

is tested to ensure that it does not shaded by any part of the tar-

get; third, valid information past the second step is imported to

the process of MoM to get E(Q) radiated from the antenna to Q;Finally, together with information like D

¼, E(Q) is substituted into

(3) to calculate total ray field received at the observation point.

Major procedure can be seen in Figure 3.

3. NUMERICAL RESULTS

In the examples given below, the working frequency is set as

300 MHz, the dimension unit of the targets is meter, and the

electrical size of the targets is set over square size of 150k2.To see the precision better, the RMS error is introduced here as

RMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼1

jrreference � rpresentedj2vuut (5)

where N is the number of data, rreference and rpresented are the

data corresponding to the referenced method and the presented

method, respectively.

The results are compared to FEKO, the commercial software

built based on MoM. Considering the computational constrain of

MoM, the examples given below are set to fulfill the extreme

computation of four common PC with parallel means.

It is quite necessary to point out that problems of arbitrary

large electrical size can be solved on one single PC by applying

the method presented in this paper.

Case 1. An arbitrary surface can be described by 16 control

points as shown in Figure 4, and a half-wave dipole is placed

above at (0.0, 0.0, 15.0) pointing to (1, 1, 1). The radiation pat-

tern of the antenna in zox-plane is calculated and shown in Fig-

ure 5 and compared to FEKO with 65,341 triangles.

The RMS error is only 0.82379 dB in this example, which

greatly shows the precision of the scheme presented. It is needed

Figure 5 Radiation pattern on zox-plane. [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com]

Figure 6 A wire helical antenna is put above the surface

Figure 7 Radiation pattern. [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. 53, No. 12, December 2011 2965

to point out that there are also direct ray field, reflected ray

field, and surface ray field contribute to the total ray field, and

obviously, the UTD method based on board, cylinder, and cone

cannot deal with this kind of surface.

In fact, for simple wire antennas like dipole and monopole,

radiation pattern from the quasihybrid scheme is much the same

as that from the analytical directional function defined in

Eq. (4), which shows the versatility of the presented scheme. If

the antenna is changed to antenna without any analytical direc-

tional function so that NURBS-UTD is limited, the quasihybrid

scheme will be more practical.

Case 2. A wire helical antenna is chosen with its radii set as

0.5, pitch set as 0.3, and height set as 0.3 to put above the sur-

face given at (0.0, 0.0, 15.0) and pointing to (0, 0, 1), as shown

in Figure 6. Radiation pattern on zox-plane and yoz-plane is cal-

culated and given in Figure 7.

The RMS error is 1.91644 dB for zox-plane and 1.71837 dB

for yoz-plane in this example, which shows that the presented

scheme can be applied to complex wire antennas without any

analytical directional function mounted around large scale sur-

face and achieve good precision.

Case 3. An aircraft constructed by 16 parametric patches is

chosen and shown in Figure 8. The helical antenna is put above

at (�45, 0.0, 9.0) and pointing to (�1, 0, 0).

The quasihybrid scheme is used to calculate radiation pattern

on yoz-plane and xoy-plane and illustrated in Figure 9. A three-

dimensional radiation pattern is also given in Figure 10. It is

needed to point out that the size of the craft is over 5000 square

wavelength, but the problem can be solved on one simple PC by

the quasihybrid scheme in NURBS-UTD method.

It can be concluded from the examples given above that

good precision can be achieved through the scheme presented,

and results are not higher than 2 dB compared to commercial

software judging by RMS error, which shows the validity; the

scheme can be applied to electrically large targets directly,

which shows the practicability.

4. CONCLUSIONS

The UTD method based on NURBS is studied in this paper. To

analyze complex wire antenna mounted around electrically large

targets while inheriting the advantages of NURBS-UTD in deal-

ing with electrically large problems, a quasihybrid scheme is

presented where MoM is used to analyze wire antenna, and

NURBS-UTD is used to analyze electrically large targets. The

information (information of reflected point, diffracted point,

etc.) resulting from ray tracing is chosen as interface between

MoM and NURBS-UTD to form the quasihybrid scheme. Sev-

eral examples are given in this paper to show the validity of the

quasihybrid scheme, and the RMS error is used to evaluate the

results and show the precision.

Figure 8 Helical antenna mounted around an aircraft

Figure 9 Radiation pattern. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]

Figure 10 Three-dimensional pattern. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com]

2966 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 DOI 10.1002/mop

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation of

China under Grant 60901030 and the Fundamental Research Funds

for the Central Universities JY10000902024.

REFERENCES

1. 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.

2. C. Ming, Z. Yu, and L. ChangHong, Calculation of the field distri-

bution near electrically large NURBS surfaces with physical-optics

method, JEMWA 19 (2005), 1511–1524.

3. K. Huang, Z.-L. He, and C.H. Liang, Improved NURBS MOM-PO

method for analyzing antenna around electrically large platform,

Microwave Opt Technol Lett 52 (2010).

4. 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.

5. C. Ming, Z. XunWang, and L. ChangHong, Analysis of antenna

around NURBS surface with iterative mom-po technique, JEMWA

20 (2006), 1667–1680.

6. Y. Haobo, W. Nan, and C. H. Liang, Fast algorithm to extract the singu-

larity of higher order moment method, JEMWA 22 (2008), 1250–1257.

7. H.B. Yuan, N. Wang, and C.H. Liang, Application of higher order

method of moments to the RCS computation of bodies modeled with

NURBS surfaces, IEEE Trans Antennas Propag 57 (2009), 3558–3563.

8. J. Perez, J.A. Saiz, and O.M. Conde, Analysis of antennas on board

arbitrary structures modeled by NURBS surfaces, IEEE Trans

Antennas Propag 45 (1997), 1045–1053.

9. W. Nan, L. Changhong, and Y. Haobo, Calculation of pattern in

UTD method based on NURBS modeling with the source on sur-

face, Microwave Opt Technol Lett 49 (2007).

10. W. Nan, Z. Yu, and C. H. Liang, Creeping ray-tracing algorithm

of UTD method based on NURBS models with the source on sur-

face, JEMWA 20 (2006), 1981–1990.

11. L. Piegl, NURBS: A survey, IEEE Computer Graph Appl (South

Florida) (1991).

12. W. Nan, Z. Yu, and L. Chang-hong, Study on reflected ray tracing

algorithm of NURBS-UTD, Chin J Radio Sci 21 (2007), 834–837.

13. W. Nan, C. H. Liang, Z. Yu, and C. Ming, Study on the creeping ray-

tracing algorithm of NURBS-UTD, J Xidian Univ 34 (2007), 600–604.

VC 2011 Wiley Periodicals, Inc.

A LOW PHASE NOISE OSCILLATOR WITHA HIGH-Q SPLIT RING RESONATORUSING MNG METAMATERIAL

Ki-Cheol Yoon and Jong-Chul Lee

Department of Wireless and Communications Engineering,Kwangwoon University, Nowon-ku, Seoul 139-701, Korea;Corresponding author: kcyoon98@kw.ac.kr or jclee@kw.ac.kr

Received 3 March 2011

ABSTRACT: In this article, a novel high QL triple-split ring resonator(T-SRR) of mu-negative metamaterial is suggested. It is used in the

design of an I- band oscillator. A very low phase noise can be obtainedfor the oscillator due to the sharp band-rejection characteristic ofT-SRR. The oscillator is designed to operate at 10 GHz using the novel

T-SRR. Experiments show that the output power is 9.82 dBm, the secondharmonic suppression is �26.48 dBc, and the phase noise is �105.05

dBc/Hz at a 100-kHz offset. VC 2011 Wiley Periodicals, Inc. Microwave

Opt Technol Lett 53:2967–2971, 2011; View this article online at

wileyonlinelibrary.com. DOI 10.1002/mop.26404

Key words: T-SRR; MNG metamaterial; high loaded quality factor; low

phase noise; oscillator

1. INTRODUCTION

In general, all modern military satellite communication systems

use oscillators for the I-band (8–10 GHz), where it is transmitted

in piloting radar using the I-band satellite [1]. There have

increasingly been requests for low phase-noise in oscillator

design, because it determines the overall performance of the sys-

tems [2]. The double-split ring resonator (D-SRR) [3] with mu-

negative (MNG) metamaterials has been reported in the design

of resonators with band-rejected performance [3]. D-SRRs are

appropriate for the oscillators and other circuits required for a

high-loaded quality factor, (QL) and band-stop resonant per-

formance due to their sharp band-rejection characteristic [3].

MNG metamaterial is artificially engineered material exhibiting

features not readily available in natural materials. The split ring

resonator (SRR), which has as its theoretical basis Veselago’s

work on metamaterial in the late 1960s [3], has engendered

great research recent interest. After Pendry’s pioneering experi-

mental work on the SRR of negative permittivity and permeabil-

ity in 1992 [4], a great deal of research has been reported in the

literature. This took advantage of the compact size and high-QL

of the novel SRR structure. This article suggests a new micro-

strip SRR resonator with high-QL and an oscillator with low

phase noise using this resonator. The proposed triple-split ring

resonator (T-SRR) has three-sections of SRRs, where the high-

QL characteristic can be obtained due to the strong coupling

effect between multiple ring structures. In addition, the resonator

size can be adjusted without variation of the resonant frequency

due to the geometric characteristics.

2. DESIGN OF A NEW HIGH-QL T-SRR WITH MNGMETAMATERIAL

2.1. High-QL T-SRR StructureThe T-SRR is composed of a triple section of rings and small

center-pad (b) connected to the inner SRR by three-short lines,

lm, as shown in Figure 1(a). From the figure, d1 and d2 are gaps

between the rings and gi (g1� g3) are gaps of the open rings. In

addition, a is the side length of the outer ring and w is the width

of the rings; these are the same in this case. Figure 1(b) shows

the equivalent T-SRR circuit. Ls is an inductance corresponding

to the total electrical length at the T-SRR. Lm is inductance due

to three-short lines at the center-pad and Cm is mutual capaci-

tance in the center-pad, used to adjust the resonant frequency.

s indicates magnetic coupling between the transmission-line and

T-SRR structure, as shown in Figure 1(a). Inductance (Ls) and

capacitances (Ci ¼ C1//C2) determine the resonant frequency

(x0) at the equivalent circuit of the T-SRR. From Figure 1(a),

the length, ln of each ring in the T-SRR is defined by the fol-

lowing Eq. (1) [5],

ln ¼ N½a� 2wðn� 1Þ � 2diðn� 1Þ� (1)

where N is the number of the ring, 3 in this case. The resonant

frequency (x0) of the T-SRR is given by (2) [6],

xo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

pr0LsCi

r(2)

where ro is the average radius of the T-SRR. The characteristic

impedances are given by (3) and (4) [7],

Z1;Zn ¼ � j

2½ðZoecotcelnÞ þ ZoocotðcolnÞ� (3)

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 2967