An equivalent network method for the analysis of nonuniform periodic structures

tive reactance contributed from the shorting pin and the w x probe; thus a wider antenna bandwidth could be achieved 4 .

The radiation patterns for the compact antenna with ls d are presented in Figure 4. It is seen that the radiation patterns remain broadside. Figure 5 shows the case for the

Ž .

antenna without slots ls 0 , where the patch resonates at about 1.923 GHz. Similar broadside radiation is also ob-served. However, because of the increase of the patch surface current component perpendicular to the main excitation

di-w x

rection 3 , the cross-polarization radiation in the H plane is

Ž .

increased cf. Figures 4 and 5 . On the other hand, the cross-polarization level in the E-plane is still in an acceptable level of less than y20 dB. Finally, it should also be noted that, due to the antenna size reduction, the antenna gain of a compact microstrip antenna will be lower than that of a conventional microstrip antenna operated at the same

fre-w x quency 2 . 3. CONCLUSIONS

The design of a compact meandered circular microstrip an-tenna with a shorting pin has been described. Experimental results have been presented and discussed. Results indicate that, by combining short-circuiting and meandering of the circular patch, the antenna size can be reduced to be less than 10% that of a conventional circular microstrip antenna operated at the same frequency. This great reduction in antenna size makes it useful for applications where antenna size is a major concern.

REFERENCES

1. K. Hiraswa and M. Haneishi, Analysis, Design, and Measurement of

Small and Low-Profile Antennas, Artech House, London, 1992.

2. R. Waterhouse, ‘‘Small Microstrip Patch Antenna,’’ Electron. Lett., Vol. 31, April 13, 1995, pp. 604]605.

3. S. Dey and R. Mittra, ‘‘Compact Microstrip Patch Antenna,’’

Microwa¨e Opt. Technol. Lett., Vol. 13, Sept. 1996, pp. 12]14.

4. T. Huynh and K. F. Lee, ‘‘Single-Layer Single-Patch Wideband Microstrip Antenna,’’ Electron. Lett., Vol. 31, Aug. 3, 1995, pp. 1310]1312.

Q 1997 John Wiley & Sons, Inc. CCC 0895-2477_r97

AN EQUIVALENT NETWORK METHOD

FOR THE ANALYSIS OF NONUNIFORM

PERIODIC STRUCTURES

Jin Jei Wu

Institute of Electro-Optical Engineering National Chiao Tung University Hsinchu, Taiwan, Republic of China

Recei¨ed 5 No¨ember 1996; re¨ised 10 February 1997

ABSTRACT: The characteristics of guided wa¨es scattered by nonuni-form wa¨eguide gratings are systematically in¨estigated with the use of an equi¨alent network method. This procedure is based on a combination of the multimode network theory and the rigorous mode-matching method. Conca¨e Bragg gratings and bent wa¨eguide gratings are taken as examples to demonstrate the present approach, and numerical results are gi¨en to illustrate their potential for millimeter-wa¨e and optical integrated circuit applications._{Q 1997 John Wiley & Sons, Inc.}

Mi-crowave Opt Technol Lett 15: 149]153, 1997.

Key words: conca¨e Bragg grating; staircase approximation

INTRODUCTION

Planar waveguide structures containing a periodic corruga-tion along the waveguide have long been used for various

w x applications, such as distributed feedback reflectors 1 ,

opti-w x

cal filters, and leaky wave antennas 2 . The electromagnetic problem of a straight uniform periodic structure has been analyzed by various methods, such as the coupled mode

w x w x

theory 1 , the finite element method 3 , the method of lines w x4 and the rigorous mode matching method 2 . With thesew x basic analyses, the properties of Bragg waveguide gratings can be considered to be well understood. In recent years, many authors have conducted work on different types of

Ž .

guided-wave gratings, such as sinusoidal or triangular pro-Ž

file gratings and tapered period interval or profile depth, .

index contrast, and so on gratings to improve and tailor the performance of active or passive devices. Thus, it is manda-tory to analyze these periodic and aperiodic structures with efficient and accurate numerical methods. The combination of the mode-matching method with multimode network the-ory has been shown to be very efficient for analyzing the nonuniform dielectric waveguides for integrated optics

appli-w x

cations 5 . In this Letter, we will utilize this combined method, which can accurately include both fundamental modes and higher-order modes, to analyze the scattering characteristics of nonuniform waveguide gratings for a mil-limeter-wave system or for optical integrated circuit applica-tions.

Numerical results presented for different types of nonuni-form Bragg gratings, such as concave waveguide gratings and bent waveguide gratings, provide guidelines for optimally designing or improving the performance of optoelectronic devices.

METHOD OF ANALYSIS

As an illustration of this method, consider a two-dimensional Ž .

structure as depicted in Figure 1 a , which shows a parallel-plate waveguide filled with a concave grating. The scattering

Ž of guided waves by nonuniform waveguide gratings such as

. concave waveguide gratings and bent waveguide gratings cannot be analyzed exactly, even though the geometric profile is simple. Thus a useful approximation is needed. We utilize the staircase approximation of the continuous nonuniform profile of the waveguide gratings; this is a discretization in

Ž .

geometry. Figure 1 b shows the staircase approximation of a concave Bragg grating in the neighborhood of x . Therefore,i

the whole structure can be approximated by a sequence of basic units, each consisting of a waveguide discontinuity and a uniform partially filled parallel-plate waveguide. According

w x

to the literature 5 , the general field solution in each uniform waveguide region can be expressed in terms of the complete

 4

set of mode functions f , which are the solutions of an

Sturm]Liouville eigenvalue problem. For the TE fundamen-tal-mode incident case, the tangential field in each uniform waveguide region can be represented by

Ž . Ž . Ž . Ž . Ez x , y s

_Ý

Ý

where V and I are the equivalent voltage and current of then n

nth mode, respectively. By matching the tangential field com-ponents at the ith discontinuity, the scattering of modes by a waveguide discontinuity can be quantified by the analysis of

Ž . Ž .

Figure 1 a Parallel-plate waveguide filled with concave Bragg gratings. b Staircase approximation of a parallel-plate waveguide Ž .

filled with concave grating. c Reflection and transmission spectrum of the fundamental TE mode, for this concave grating with top

'

and bottom cover: Rs 200 cm, H s 0.54 cm, t s 0.045 cm, t s t s 0.09 cm, n s 5 , n s 1.0, L s 0.36 cm, and 20 periodsg b m d a 0

modal currents and voltages. For the voltages and currents on the two sides of the ith step, we obtain the linear systems:

Ž y. Ž . Ž q. Ž . Vm xi s

Ý

_Ý

where Qi mnis an element of the coupling matrix at the ith step discontinuity between uniform waveguides. The

ele-ments of this matrix can be defined as the scalar products of the mode functions on the two sides of this discontinuity, as follows:

ŽQi.mns²f xmŽ yi .Nf xnŽ qi .:. Ž .5 Note that

ŽQiy1.mns QŽ i.mn, so that Qy1i s Q ,ti

Ž . Ž .

From 5] 7 it can be easily deduced that the input

Ž y. y

impedance matrix Z xi at the xs x plane looking to thei

right satisfies

Ž y. Ž Ž q. . t Ž .

Z xi s Q Z xi i Q ,i 6

where the Q matrix is previously defined. From the impedance Ž .

transform matrix formula 6 the reflection coefficient matrix

Ž y. y Ž q .

G x at the x s x plane and the impedance matrix Z xi i iy1

at the xs xqiy1 plane looking to the right can be obtained as

y1 y y y Ž . w Ž . x w Ž . x Ž . G xi s Z xi q Zoi Z xi y Z ,oi 7 and y1 q Ž . w xw x Ž . Z xiy1 s Z I q H G H I y H G Ho i i i i i i i , 8 where Zoi and H are the characteristic impedance matrixi

and the phase matrix of the ith step discontinuity; their elements are defined as

ŽZoi.mnsd Z ,mn oi n Ž .9 ŽHoi.mnsdmnexpŽyjк lx i n i. Ž10. where кx i n and Zoi n are, respectively, the wave number in

the x direction and the characteristic impedance for the nth mode in the ith dielectric waveguide section with length l . Iti

is obvious that the staircase approximation is very simple, and can be generally applied to nonuniform integrated optical devices with any profiles.

NUMERICAL RESULTS

To illustrate the electromagnetic scattering characteristics of nonuniform periodic structures, we present the following results. Consider the first case of a parallel-plate waveguide

Ž . filled with a concave Bragg grating, shown in Figure 1 a . The profile of the concave grating region is characterized by a circle. We assume that each uniform dielectric waveguide region of the grating supports only the TE guided mode. For0

Ž . the mode-matching procedures, the number of modes M , including propagating and evanescent modes, has been set at Ž . 15 throughout this article for numerical analysis. Figure 1 c shows the normalized reflection and transmission power as a function of wavelength l for the structure of a parallel-plate waveguide filled with a concave Bragg grating, as shown in

Ž .

Figure 1 a with Ns 20 periods. The results show that there exist a main stop band and a series of tapered-intensity side lobes extending to smaller l. We find, if the radius of curvature of the concave region is large enough, that the central frequency of the main stop band can be roughly

Ž . Ž .

Figure 2 a Structure configuration of a Bragg grating bent upward along the parabola. b Structure configuration of a Bragg grating Ž .

bent downward along with parabola. c Reflection and transmission spectrum of the fundamental TE mode of bent waveguide gratings Ž .

bending with parabola curves: nds 1.5, n s 1.0, t s 0.36a f mm, t s 0.14 mm, L s 0.6232 mm and 25 periods. d Comparison ofg 0

estimated from the uniform-grating Bragg condition, and that there exists a wide passband belowl s 0.97 cm.

In order to analyze open types of nonuniform periodic structures for integrated optics applications with the method

w Ž .x

presented here, we extend the two metal plates Figure 1 a upwards and downwards, respectively, so as to minimize the effect on the grating caused by these plates. We preserve the metal plates in order to reduce the continuous spectrum of radiation modes into a complete set of discrete modes, as is

w x

customarily done in the literature 5 . Thus, part of the power of the TE surface mode, which by scattering at the disconti-0

nuities of the open-type waveguide grating is coupled into the higher-order modes, can be viewed as the radiation associ-ated with the continuous spectrum. The open nonuniform

periodic gratings analyzed here are bent-waveguide-type grat-ings. The examples that we will discuss first are Bragg wave-guide gratings bent into a parabolic shape, as shown in

Ž . Ž .

Figures 2 a and 2 b . We use the notation qh for the displacement of the top point of the upward parabola wave-guide grating above the horizontal direction of the

Ž .

inputroutput waveguide shown in Figure 2 a ; yh represents displacement of the bottom of the downward-bent waveguide

Ž .

grating shown in Figure 2 b . The numerical results for the two types of bending are compared to that of the flat one in

Ž . < <

Figure 2 c . Apparently, if the value of h increases gradually and the grating is bent more severely, the side lobes of the reflection spectrum decrease for a downward-bent waveguide grating, whereas an upward-bent grating shows the opposite

Ž . Ž . ya x Ž .

Figure 3 a Structure configuration of a bent waveguide grating with a twist and bending along with y x s e symmetrically. b Ž .

Reflection and transmission spectrum as a function of l for the structure of a : n s 3.4, n s 1.0, t s 0.15 mm, t s 0.06 mm,d a f g

Ž .

L s 0.3040 mm, and 25 periodics. c Comparison of the variation of backward and forward radiation between uniform grating and Ž .

effect. The bandwidth of the stop band changed slightly. For the downward parabolic bending of the grating, the side lobes

Ž

are reduced, so part of the energy depending on the struc-.

ture parameters and incident wavelength l will be released Ž .

as radiation loss. Figure 2 d shows the comparison of the backward and forward radiation powers between the straight and bent waveguide gratings. The forward radiation means that the energy is radiated from grating in the same lateral direction as the incident wave, and the backward radiation is in the opposite direction. For uniform waveguide grating, there exists a backward leaky wave in the range of l s 1.3]1.42 mm. The existence of the leaky region arises from the periodic nature of the grating waveguide; this has been

w x

pointed out in the literature 6 . The guided-wave scattering by the finite periodic discontinuities of grating will lead to more energy leakage into the air region if the average propa-gation constant b in the waveguide grating satisfies thed

inequality

Ž . Ž .

kaGb s b q 2 nprL ,n d 0 n s 0, " 1, " 2 ??? 1 where ka is the propagation constant in air and L is the0

grating periodicity. For the leaky-wave region indicated in Ž .

Figure 2 d , the only value that could satisfy the leakage Ž .

condiction 11 is n s y1. In this case, b_y1 is negative, so that the field scattered in the air region is inclined toward the yx direction. The nonuniformity of the guiding structure also leads to more radiation loss into exterior region. There-fore, the radiation loss in the leaky-wave region will increase by bending the waveguide grating upward. Thus, the bent waveguide grating mentioned above has some advantages: For an upward-bending grating the radiation loss in the leaky-wave region will increase and may enhance the perfor-mance of a leaky-wave antenna. For a downward-bending grating, side lobes will be more efficiently suppressed and the grating can be used as a distributed feedback reflector. These conclusions hold also for optical material of a higher

refrac-Ž .

tion index for example, nds 2.0, 3.4, and so on . Besides, if

there exists a drastic twist at the bottom of the bent wave-Ž .

guide grating, such as indicated in Figure 3 a , the reflection and transmission coefficient will also be affected. Such a twist may occur in waveguide grating because of a manufacturing imperfection. The waveguide grating under discussion bends down to and then up from its minimum, with the bend having

Ž .

an exponential profile. Figure 3 b shows the normalized reflection and transmission powers as a function of

wave-Ž .

length, and Figure 3 c shows the forward and backward radiation powers. For comparison, we also present the data for the uniform grating, shown by solid lines in these two figures. The numerical results show that if there is a twist, the stop-band intensity drops somewhat, its bandwidth narrows, and the side lobes nearest the primary reflection peak will increase. It is also found that forward radiation will increase

Ž . by bending the waveguide grating, as shown in Figure 3 a . In summary, an accurate analysis of the effects caused by

Ž .

nonuniform i.e., bent gratings can be used effectively to manipulate the electromagnetic scattering characteristics of these waveguide gratings. Nonuniformities in waveguide grat-ings may provide an extra degree of freedom in designing and improving the performance of devices.

CONCLUSION

We have analyzed five types of nonuniform Bragg gratings by staircase approximation. Numerical results show that

scatter-ing properties of these nonuniform Bragg gratscatter-ings can be Ž determined by the shape of the nonuniformities such as

.

bending functions . Therefore, these types of periodic struc-tures may provide another degree of freedom to design waveguide gratings for millimeter-wave or optical-frequency application.

REFERENCES

1. H. Kogelnik and C. V. Shank, ‘‘Coupled-Wave Theory of Dis-tributed Feedback Lasers,’’ J. Appl. Phys., Vol. 43, No. 5, 1972, pp. 2328]2335.

2. S. T. Peng, ‘‘Rigorous Formulation of Dielectric Grating Wave-guides}General Case of Oblique Incidence,’’ J. Opt. Soc. Am.

Ser. A, Vol. 6, 1989, pp. 1869]1883.

3. S. J. Chung and Jiunn-Lang Chen, ‘‘A Modified Finite Element Method for Analysis of Finite Periodic Structures,’’ IEEE Trans.

Microwa¨e Theory Tech., Vol. MTT-42, July 1994, pp. 1561]1566.

4. R. Pregla and W. Yang, ‘‘Method of Line for Analysis of Multilay-ered Dielectric Waveguides with Bragg Gratings,’’ Electron Lett., Vol. 129, No. 22, 1993, pp. 1962]1963.

5. S. J. Xu, S. T. Peng, and F. K. Schwering, ‘‘Transition in Open Millimeter-Wave Waveguides,’’ IEE Proc. Pt. H, Vol. 136, No. 6, 1989, pp. 487]491.

Ž .

6. T. Tamir Ed. , Integrated Optics, Springer-Verlag, Berlin, 1975, pp. 96_]100.

Q 1997 John Wiley & Sons, Inc. CCC 0895-2477r97

DIPOLE ANTENNAS ON PHOTONIC

BAND-GAP CRYSTALS

} EXPERIMENT

AND SIMULATION

M. M. Sigalas,1_{R. Biswas,}1_{Q. Li,}1_{D. Crouch,}2_{W. Leung,}3

Russ Jacobs-Woodbury,3Brian Lough,3Sam Nielsen,3 S. McCalmont,3_{G. Tuttle,}3_{and K. M. Ho}3

1_{Ames Laboratory}

Microelectronics Research Center Department of Physics and Astronomy Iowa State University

Ames, Iowa 50011

2

Hughes Electronic Corporation AET Center

P.O. Box 1973

Rancho Cucamonga, California 91729

3

Ames Laboratory

Microelectronics Research Center Iowa State University

Ames, Iowa, 50011

Recei¨ed 20 January 1997

ABSTRACT: The radiation patterns of dipole antennas on

three-dimen-sional photonic crystal substrates ha¨e been measured and calculated with the finite-difference]time-domain method. The photonic band-gap crystal beha¨es as a perfectly reflecting substrate, and all the dipole power is radiated into the air side when dri¨en at frequencies in the stop band. The radiation pattern is found for different positions and orientations of the dipole antenna. Antenna configurations with desirable patterns are identified.Q 1997 John Wiley & Sons, Inc. Microwave Opt Technol

Lett 15: 153]158, 1997.

Key words: photonic band-gap materials; dipole antennas;