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

Žare 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 thebackward and forward radiation powers between the straightand bent waveguide gratings. The forward radiation meansthat the energy is radiated from grating in the same lateraldirection as the incident wave, and the backward radiation isin the opposite direction. For uniform waveguide grating,there exists a backward leaky wave in the range of l s1.3]1.42 mm. The existence of the leaky region arises fromthe periodic nature of the grating waveguide; this has been

w xpointed out in the literature 6 . The guided-wave scatteringby the finite periodic discontinuities of grating will lead tomore energy leakage into the air region if the average propa-gation constant b in the waveguide grating satisfies thedinequality

Ž . Ž .k G b s b q 2 nprL , n s 0, " 1, " 2 ??? 1a n d 0

where k is the propagation constant in air and L is thea 0grating periodicity. For the leaky-wave region indicated in

Ž .Figure 2 d , the only value that could satisfy the leakageŽ .condiction 11 is n s y1. In this case, b is negative, soy1

that the field scattered in the air region is inclined toward theyx direction. The nonuniformity of the guiding structurealso leads to more radiation loss into exterior region. There-fore, the radiation loss in the leaky-wave region will increaseby bending the waveguide grating upward. Thus, the bentwaveguide grating mentioned above has some advantages:For an upward-bending grating the radiation loss in theleaky-wave region will increase and may enhance the perfor-mance of a leaky-wave antenna. For a downward-bendinggrating, side lobes will be more efficiently suppressed and thegrating can be used as a distributed feedback reflector. Theseconclusions hold also for optical material of a higher refrac-

Ž .tion index for example, n s 2.0, 3.4, and so on . Besides, ifdthere exists a drastic twist at the bottom of the bent wave-

Ž .guide grating, such as indicated in Figure 3 a , the reflectionand transmission coefficient will also be affected. Such a twistmay occur in waveguide grating because of a manufacturingimperfection. The waveguide grating under discussion bendsdown to and then up from its minimum, with the bend having

Ž .an exponential profile. Figure 3 b shows the normalizedreflection and transmission powers as a function of wave-

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

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

Ž .nonuniform i.e., bent gratings can be used effectively tomanipulate the electromagnetic scattering characteristics ofthese waveguide gratings. Nonuniformities in waveguide grat-ings may provide an extra degree of freedom in designing andimproving the performance of devices.

CONCLUSION

We have analyzed five types of nonuniform Bragg gratings bystaircase approximation. Numerical results show that scatter-

ing properties of these nonuniform Bragg gratings can beŽdetermined by the shape of the nonuniformities such as

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

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 ElementMethod 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 OpenMillimeter-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 PHOTONICBAND-GAP CRYSTALS } EXPERIMENTAND SIMULATIONM. M. Sigalas,1 R. Biswas,1 Q. Li,1 D. Crouch,2 W. Leung,3

Russ Jacobs-Woodbury,3 Brian Lough,3 Sam Nielsen,3

S. McCalmont,3 G. Tuttle,3 and K. M. Ho31 Ames LaboratoryMicroelectronics Research CenterDepartment of Physics and AstronomyIowa State UniversityAmes, Iowa 500112 Hughes Electronic CorporationAET CenterP.O. Box 1973Rancho Cucamonga, California 917293 Ames LaboratoryMicroelectronics Research CenterIowa State UniversityAmes, 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 calculatedwith the finite-difference]time-domain method. The photonic band-gapcrystal beha¨es as a perfectly reflecting substrate, and all the dipole poweris radiated into the air side when dri en at frequencies in the stop band.The radiation pattern is found for different positions and orientations ofthe dipole antenna. Antenna configurations with desirable patterns areidentified. Q 1997 John Wiley & Sons, Inc. Microwave Opt TechnolLett 15: 153]158, 1997.

Key words: photonic band-gap materials; dipole antennas; finite-dif-ference]time-domain calculations

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 153

INTRODUCTION

A novel class of periodic dielectric structures has been devel-Ž .oped. In this class, propagation of electromagnetic EM

waves is forbidden for all frequencies in the stop band orŽ . w xphotonic band gap PBG 1, 2 . These PBG structures have

been fabricated at a variety of microwave and millimeter-wavelength scales with three-dimensional PBG frequencies rang-

w xing between 10 and 500 GHz 2]5 . An extremely attractiveapplication of these PBG crystals is as reflecting substratesfor integrated circuit antennas.

A serious disadvantage of conventional antennas on aŽsemiinfinite semiconductor substrate with dielectric constant

.e is that the power radiated into the substrate is a factor3r2 w xe larger than the power into the free space 6 . Hence,

antennas on GaAs or Si radiate only 2]3% of their powerinto free space. A large fraction of the power radiated intothe substrate is in the form of trapped waves propagating at

w xangles larger than the critical angle 6 . By fabricating theantenna on a PBG material with a driving frequency in thestop band, no power should be transmitted into the PBGmaterial and all power should be radiated in the free space,provided there are no surface modes.

w xBrown, Parker, and Yablonovitch 7 demonstrated thisconcept by fabricating a bow-tie antenna on their three-cylin-der PBG crystal and found a complex radiation pattern in air.They improved the directionality by placing the antenna on

w xdifferent high- and low-dielectric surfaces 8, 9 . Cheng et al.w x10 measured the dipole radiation pattern on the layer-by-layer PBG crystal and found the pattern to be stronglydependent on the position of the dipole in the unit cell.

w xKesler, Maloney, Shirley, and Smith 11 measured the radia-Ž .tion of antennas placed on top of two-dimensional 2D and

Ž .three-dimensional 3D PBG materials and found insensitiv-ity of the antenna patterns when the dipole was raisedsufficiently high above the surface.

In this article we measure and calculate the radiationpattern of a dipole antenna on our layer-by-layer PBG crystalw x3 . The layer-by-layer PBG crystal was fabricated by stackingalumina rods. It has a three-dimensional stop-band gap be-tween 12 and 14 GHz with an attenuation of more than 40

w xdB within the stop band 3 . The four variables controllingŽ .the antenna radiation pattern are a the position of the

Ž .antenna in the surface unit cell, b the height of the antennaŽ . Ž .above the surface, c the orientation of the antenna, and d

the driving frequency.There are two high symmetry positions in the unit cell

Ž .corresponding to the antenna: a in a solid position on top ofthe rod, that is, intersection of the first- and second-layer

Ž .rods, or b in the void position with no dielectric rod directlyŽbeneath it, but above the third- and fourth-layer rods Figure

.1 . At each of these two surface positions there are twoorientations, corresponding to the dipole parallel or perpen-dicular to the first-layer rods. This implies a total of fourdifferent configurations for the antenna in the PBG unit cell.For each configuration the antenna radiation pattern wasmeasured as a function of height z above the surface, and

Ž .compared with finite-difference]time-domain FDTD calcu-lations. The height z is expressed as a ratio of the rod

Ž .diameter d d s 0.318 cm . Other lower symmetry positionsin the unit cell were not found to have any particular advan-tage. Experimentally, the minimum height of the dipole abovethe top of the first-layer rods is the coaxial feed cable radius,corresponding to a minimum height of zrd , 0.53.

Figure 1 Schematic geometry of the surface of the PBG crystalviewed along the stacking direction. Rods of the first four layers areshown, and the solid and void positions in the unit cell are indicated.The dipole antenna can be oriented either parallel or perpendicularto the first layer, resulting in the four configurations of the dipole

In experiments a Ku-band synthesizer generated an inputsignal that was divided by a 3-dB hybrid coupler into twocomponents that were 1808 out of phase. Each componentsignal was routed through adjustable phase shifters and 50-V

w xcoaxial cables 10 . The dipole was fabricated by bending thecenter conductors of the two coaxial cables and minimizingthe feed gap. Measurements were performed in an anechoicchamber with an HP8510B network analyzer, with the dipoleas a rotating source and the pyramidal feed horn as thereceiving antenna. Feed wires rotated with the dipole, tominimize electromagnetic interference. Exceptional care infabrication was used so that the radiation of the free dipolewas found to be very close to the expected result.

Antenna radiation patterns were simulated with the widelyŽ . w xused finite-difference]time-domain FDTD technique 12

by numerically integrating the time-dependent Maxwell’sequations for a dipole source on the PBG crystal. The FDTD

Žcalculations were performed for the finite-length dipole 1.6.cm used in the measurements. The symmetry of the crystal

reduced the FDTD computational space to one-fourth of theactual PBG crystal cell. Both calculations and measurementsare shown at 13 GHz at the center of the photonic band gap,

Ž .and results are not sensitive to small f 0.5 GHz frequencyvariation.

RESULTS

( )2 Perpendicular Solid Position. For the dipole above theŽ .first-layer rod, at its minimum height zrd s 0.53 , both

experiment and FDTD calculations predict a central lobe inthe E plane and a broader but still centrally peaked pattern

Ž . Žin the H plane Figure 2 . As the dipole is raised zrd s.1.3]1.5 the E-plane intensity weakens, with two side lobes

developing at about 408 to the normal. The H-plane pattern

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997154

Ž . Ž .Figure 2 Measured a and FDTD calculations b of the antenna radiation in the E and H planes when the dipole antenna is in theperpendicular solid position, for a frequency of 13 GHz at the center of the photonic band gap. The three sets of curves correspond todifferent heights z of the antenna above the top of the first-layer rod. Heights z are expressed as a ratio zrd, where d is the diameter

Ž .of the dielectric rod 0.318 cm . Experimental uncertainty in zrd is approximately "0.1. The intensity units are arbitrary

weakens but remains broad. As the dipole is lifted higherŽ .zrd s 2.5 , both the E- and H-plane intensities decreasefurther, with a central lobe and two side lobes at about 408 tothe normal. The weakening radiation arises from a phasecancellation between the direct wave and the wave reflectedfrom the photonic crystal. There is very good agreementbetween experiment and FDTD calculation. It is desirable toplace the dipole close to the surface to achieve strong centralpeak patterns.

( )2 Parallel Solid Position. Experimentally the pattern is acentral lobe in E and H planes when the dipole is close to the

Ž . Ž .surface 0.5 - zrd - 1.5 Figure 3 . The H plane is consid-Ž .erably broader than the E plane, as in position 1 . However,

Ž .when the dipole is raised zrd s 1.3]1.5 the radiated inten-sity increases. This strengthening of radiated power isoveremphasized in the FDTD calculations compared to ex-

Ž .periment. For larger heights zrd s 2.5 the H plane devel-ops a strong two-side-lobe pattern, and weak side lobesappear in the E-plane pattern as well. It may be advanta-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 155

Figure 3 Measurements and FDTD calculations for the antenna in the parallel solid position. The same conventions as in Figure 2are followed

geous to stay close to the surface but lift the dipole slightly toobtain the strongest central lobe pattern. The strong side

Ž .lobes of radiated power when the dipole is raised zrd G 2.5may be useful for directional antennas.

( )3 Perpendicular Void Position. The E plane is a broad peak,whereas the H plane is a broad distribution spanning 608about the normal when the dipole is close to the surfaceŽ . Ž . Žzrd - 0.5 Figure 4 . When the dipole is raised zrd s

.1.3]1.5 , weak side lobes in the E plane and two strong sidelobes at 608 to the normal in the H plane appear in boththeory and measurement. As the dipole is raised furtherŽ .zrd , 3 the side lobes in the H plane weaken and astronger central lobe appears, and the E plane also developsa central lobe. In fact, when the antenna is raised even

Ž .further zrd s 4, not shown , the E and H planes havestrong central lobes. To achieve central lobes the antenna

Ž .needs to be raised considerably zrd s 4, not shown . Other-wise this position leads to strong directional lobes in the Hplane for the antenna close to the surface. This position maynot be desirable for antenna placement unless the directionallobes in the H plane can be utilized.

( )4 Parallel Void Position. The patterns display the sameŽ .trends as the parallel solid position 2 , and are not shown

separately. It is necessary to have the antenna close to theŽ .surface 0.5 - zrd - 2 for narrow central peaks in the E

plane and broad peaks in the H plane. Raising the antennaabove zrd s 2 produces side lobes.

We find that that antenna patterns become insensitive to

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997156

Figure 4 Measurements and FDTD calculations for the antenna in the perpendicular void position. The same conventions as inFigure 2 are followed

position when the antenna is raised high enough above theŽ .substrate e.g., zrd ) 3 , where the E fields are uniform, and

results for void and solid positions are similar.w xThe earlier measurements on a printed-circuit dipole 10

had radiation patterns that were different from that expectedfor the free dipole, leading to differences between the pre-sent measurements and that of the printed-circuit dipole. Theheight dependence of the patterns was not previously investi-

w xgated 10 . Present measurements for the perpendicular solidw xposition are in good agreement with the previous results 10 .

In all positions there is virtually no power propagatingthrough the PBG crystal with all radiation emerging from thefront side, illustrating the PBG crystal behaving as a perfectdissipationless reflector.

CONCLUSIONS

The radiation pattern of a dipole antenna on the surface of aPBG crystal was measured and calculated with the FDTDmethod for different surface positions, heights, and orienta-tions of the dipole. Within the stop band the PBG crystalbehaves as a dissipationless reflector and all the antennapower is reflected into the air side.

The antenna radiation strongly depends on the position ofthe dipole antenna, because the dipole radiation depends onthe rapidly varying local E field at the surface of the photoniccrystal. When the dipole is raised high enough above thesurface the patterns become insensitive to dipole position,because the E field is uniform. For our layer-by-layer PBGcrystal we find the two dipole orientations directly on top of

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 157

the dielectric rod to be the most suitable for central lobe typeof patterns. It is advantageous to have the dipole placeddirectly above a dielectric. At several geometries when thedipole is raised above the surface we find strong lobes in theH plane that may be useful for directional antennas.

There is good agreement between the measurements andthe FDTD calculations, especially in the angular dependenceof the patterns. FDTD simulations can be used as a powerfuldesign tool to optimize antenna patterns on any photoniccrystal surface. Antennas in cavity geometries may be de-signed on PBG crystal surfaces, to optimize the antennaradiation. PBG crystals have immense potential for novelmicrowave and millimeter-wave applications.

ACKNOWLEDGMENTS

The authors would like to thank R. Weber and C. M. Souk-oulis for helpful discussions. Ames Laboratory is operated bythe U.S. Department of Energy by Iowa State Universityunder Contract No. W-7405-Eng-82. We also acknowledgesupport by the Department of Commerce through the Center

Ž .of Advanced Technology Development CATD at Iowa StateUniversity

REFERENCESŽ .1. C. M. Soukoulos Ed. , Photonic Band Gap Materials, Proc. NATO

ASI, Plenum, New York, 1996.2. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, ‘‘Photonic

Band Structure: The fcc Case Employing Non-Spherical Atoms,’’Phys. Re¨. Lett., Vol. 67, Oct. 1991, pp. 2295]2298.

3. E. Ozbay, A. Abeyta, G. Tuttle, M. Tringides, R. Biswas, C. T.Chan, C. M. Soukoulis, and K. M. Ho, ‘‘Measurements of aThree Dimensional Photonic Band Gap in a Crystal StructureMade of Dielectric Rods,’’ Phys. Re¨. B, Vol. 50, July 1994, pp.1945]1948.

4. E. Ozbay, E. Michel, G. Tuttle, M. Sigalas, R. Biswas, and K. M.Ho, ‘‘Micromachined Millimeter Wave Photonic Band Gap Crys-tals,’’ Appl. Phys. Lett., Vol. 64, April 1994, pp. 2059]2061.

5. E. Ozbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak,and D. M. Bloom, ‘‘Terahertz Spectroscopy of Three-Dimen-sional Photonic Band Gap Crystals,’’ Opt. Lett., Vol. 19, Aug.1994, pp. 1155]1159.

6. D. B. Rutledge, D. P. Neikirk, and D. P. Kasilingam, in Infraredand Millimeter Wa¨es Academic, Orlando, 1983, Vol. 10, p. 1.

7. E. R. Brown, C. D. Parker, and E. J. Yablonovitch, ‘‘RadiationProperties of a Planar Antenna on a Photonic Crystal Substrate,’’J. Opt. Soc. Am. Ser. B, Vol. 10, Feb. 1993, pp. 404]407.

8. E. R. Brown, C. D. Parker, and O. B. McMahon, ‘‘Effect ofSurface Composition on the Radiation Pattern from a PhotonicPlanar Dipole Antenna,’’ Appl. Phys. Lett., Vol. 64, June 1994,pp. 3345]3347.

9. E. R. Brown and O. B. McMahon, ‘‘High Zenithal Directivityfrom a Dipole Antenna on a Photonic Crystal,’’ Appl. Phys. Lett.,Vol. 68, June 1994, pp. 1300]1302.

10. S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, andK.-M. Ho, ‘‘Optimized Dipole Antennas on Photonic Band GapCrystals,’’ Appl. Phys. Lett., Vol. 67, Dec. 1995, pp. 3399]3401.

11. M. P. Kesler, J. G. Maloney, B. L. Shirley, and G. S. Smith,‘‘Antenna Design with the Use of Photonic Band Gap Materialsas All Dielectric Planar Reflectors,’’ Microwa e Opt. Technol.Lett., Vol. 11, March 1996, pp. 169]174.

12. K. Umashankar and A. Taflove, Computational Electromagnetics,Artech House, Boston, 1993.

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

THE PSTD ALGORITHM: ATIME-DOMAIN METHOD REQUIRINGONLY TWO CELLS PER WAVELENGTH

Q. H. LiuKlipsch School of Electrical and Computer EngineeringNew Mexico State UniversityLas Cruces, New Mexico 88003

Recei ed 20 January 1997

( )ABSTRACT: A pseudospectral time-domain PSTD method is de¨el-oped for solutions of Maxwell’s equations. It uses the fast Fourier

( )transform FFT , instead of finite differences in con¨entional finite-dif-( )ference]time-domain FDTD methods, to represent spatial deri ati es.

Because the Fourier transform has an infinite order of accuracy, only twocells per wa¨elength are required, compared to 8]16 cells per wa¨elengthrequired by the FDTD method for the same accuracy. The wraparoundeffect, a major limitation caused by the periodicity assumed in the FFT,is remo¨ed by using Berenger’s perfectly matched layers. The PSTD

D D (method is a factor of 4 ]8 more efficient than the FDTD methods D)is the dimensionality . Q 1997 John Wiley & Sons, Inc. Microwave Opt

Technol Lett 15: 158]165, 1997.

( )Key words: Pseudospectral time-domain PSTD method; perfectly( )matched layers PML ; conducti e medium; inhomogeneous medium;

( )transient wa¨e scattering; finite-difference]time-domain FDTD method.

I. INTRODUCTION

Ž .Finite-difference]time-domain FDTD methods are widelyapplied to simulate transient electromagnetic wave propaga-tion. In these methods, the spatial derivatives in Maxwell’sequations are approximated by finite differences. In the most

w xpopular FDTD method, the Yee algorithm 1 , the electricand magnetic field components are located at different posi-tions in a staggered grid. The Yee algorithm has a second-order accuracy in both space and time. Therefore, it givesvery satisfactory results if the discretization is fine enough.Numerical experiments show that accurate results require afine discretization of about 10]20 cells per wavelength at thehighest frequency being simulated.

To increase the efficiency of time-domain solutions byreducing the grid density, higher-order finite differences canbe used. The spectral-domain methods can be viewed as afinite-difference method with an infinite order of accuracy.There are two basic formulations for spectral-domain meth-

Ž .ods: The generalized k space GKS method solves the inte-w xgral equation derivable from Maxwell’s equation 2]4 , and

the pseudospectral method solves the original partial differ-w xential equations 5]8 . In the spectral-domain methods, be-

cause the Fourier transforms used to represent the spatialderivatives or integrals are exact, there is no dispersion error.Numerical examples have been shown to demonstrate the

w xhigh efficiency of the spectral-domain methods 3, 4, 7, 8 .In spectral-domain methods, the discrete Fourier trans-

Ž .form DFT is achieved by using the fast-Fourier-transformŽ .FFT algorithm. Therefore, unlike the FDTD methods, whichrequire 10]20 cells per wavelength, the spectral-domainmethods require only 2 cells per wavelength, which is dictatedby the Nyquist sampling theorem. Hence, these methods aremuch more efficient than FDTD methods, especially forproblems with large scatterers. However, because the spec-tral-domain methods use DFT, a periodicity is assumed inspatial dimensions. The late-time solutions are therefore cor-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997158