proportional to the input pulse width; thus, the compression ratioremains the same. The compression is therefore Doppler invariant.

The main advantage of the matched filter method based on ahyperbolic frequency modulation is that it can achieve very highcompression ratio. The compression ratio for the compression of achirp is given by

C � TW, (20)

where W is the effective bandwidth and T is the duration of thepulse. Normally the chirp bandwidth is large and the compressionratio is more than 100. If the bipolar pulse and a multiple integratorare used to achieve the same degree of compression, the order ofintegration will be as large as several thousands, based on Eq. (19).This is simply impossible because the peak value of the output willbe extremely small, according to Eq. (18). Another advantage ofthe matched filter method is the noise performance is much better,because the noise components are not matched to the filter and willbe filtered out.

5. CONCLUSION

A hyperbolic frequency-modulated waveform is Doppler-invariantif the relative velocity between the radar and the target is constant.Numerical examples indicate that a linear frequency-modulatedpulse has large degradation in performance when the Dopplereffect is large. The pulse width is enlarged and the peak value isreduced significantly. While the result of the hyperbolic frequency-modulated pulse is still a desired narrow pulse, which is verysimilar to the result without the Doppler effect. Using hyperbolicfrequency-modulated waveform and a matched filter a large com-pression ratio can be achieved, compared to another Doppler-invariant pulse compressor based on the bipolar waveform and themultiple integrator. The better performance of the hyperbolic fre-quency modulation under Doppler distortion makes it very usefulin detecting fast-moving objects.

REFERENCES

1. I. Faulconbridge, Radar fundamentals, Argos Press, Australia, 2002.2. J.J. Kroszczynski, Pulse Compression by means of linear-period mod-

ulation, Proc IEEE 57 (1969), 1260–1266.3. R.A. Altes and E.L. Titlebaum, Bat signals as optimally Doppler tol-

erant waveforms, J Acoust Soc 48 (1970), 1014–1020.4. E.J. Kelly and R.P. Wishner, Matched-filter theory for high-velocity,

accelerating targets, IEEE Trans Mil Electron MIL-9 (1965), 56–65.5. D. Hazony and Y. Hazony, Doppler invariant pulse compressors, J

Franklin Inst 309 (1980), 215–225.

© 2006 Wiley Periodicals, Inc.

SELECTIVE RE-MESHING: A NEWAPPROACH TO INCLUDE MODESTIRRING EFFECTS IN THE STEADYSTATE FDTD SIMULATION OFMICROWAVE HEATING CAVITIES

Jacob George and Richard BergmanScience and Technology DivisionCorning IncorporatedCorning, NY 14831

Received 7 December 2005

ABSTRACT: A new selective remeshing approach to efficiently utilizethe finite-difference time-domain (FDTD) method for simulating 3D mi-crowave heating cavities with rotating mode stirrers is presented here.The proposed scheme is implemented in a �300L multimode applicatorand the result compares favorably with a validation experiment. The pro-posed implementation of the selective remeshing scheme is efficient, in thatthe remeshing is done “on the fly” and the previous FDTD solution is usedto continue the simulation as a single run of the FDTD solver. © 2006Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1179–1182, 2006;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.21572

Key words: FDTD; microwave heating; mode stirrer

1. INTRODUCTION

Microwave heating is widely used in industry to achieve faster,cheaper, and efficient heating of dielectric materials. It is experi-mentally well known that mode stirrers are beneficial in terms ofimproving heating uniformities inside microwave cavities [1] andare widely used in industrial applicators. Modeling and simulationof the effects of mode stirrers inside typical microwave heatingcavities has always been challenging. This has led to applicatordesigns based on experience and empiricism, rather than predictivecomputer simulation. There are few reports in the literature [2, 3]regarding efforts towards estimating the effect of these movingparts on the power dissipation inside the dielectric materials. Thereported methods [2, 3] use the 2D finite-element method (FEM)to predict the effect of laminate mode stirrers on the heatingpattern inside the processed material. Despite the heavy use of the3D FDTD method as a tool for predicting and understandingmicrowave-assisted thermal processing of materials [4–6, 8], therehas not been any detailed effort to apply it to rotating mode stirrers(in contrast to the laminate mode stirrers described in [2, 3]) inheating cavities.

In this paper, we present a 3D FDTD-based approach, whichutilizes a novel and efficient selective remeshing technique, forcomputing the steady-state power-dissipation pattern inside lossydielectric materials during microwave-assisted thermal processingin a cavity with a rotating mode stirrer. A qualitative experimentalvalidation of the proposed approach is also presented.

2. SELECTIVE REMESHING APPROACH

The selective remeshing approach is based on the assumption thatthe heating rate of the material to be processed is slow comparedto mode-stirrer rotation [1–3]. This assumption enables us to solvefor the electromagnetic fields inside the material to be processedby running multiple steady-state FDTD simulations correspondingto different angular orientations of the mode stirrer.

According to this new approach, first the meshing of the cavityto be simulated is done by specifying a uniform grid of Yee’s cells[7] throughout the domain, specifying the spatially dependent

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dielectric properties. Then a conventional FDTD algorithm [7] isrun until the system reaches steady state [8]. At steady state, threethings are done: (i) the power-dissipation field inside the dielectricload is calculated and saved; (ii) the remeshing step is done byreassigning the dielectric properties of the underlying mode stirrergeometry to the fixed grid in order to simulate the updated modestirrer position; (iii) the FDTD solver is resumed from the previ-ously attained steady state solution. Steps 1–3 are repeated untilsteady-state solutions for all the different angular positions of themode stirrer are obtained.

This new approach for solving a microwave heating cavityproblem is demonstrated with an example that includes a three-leafrotating-mode stirrer.

3. NUMERICAL AND EXPERIMENTAL DETAILS ANDRESULTS

The selective remeshing approach is applied to a �300L cavity, asshown in Figure 1(a). The cavity contains a rotating mode stirrerwhose blades interact at different angular positions relative to theWR-340 waveguide feed entrance in the back of the applicator, asshown in Figure 1(b). Materials A (insulation, with �r � 1.7 �

j0.001) and B (laminated fiberglass [1] equivalent, with �r �4.4 � j0.128) are simulated as lossy dielectrics. All the metallicparts in the cavity are considered as perfect electric conductors(PEC) in the simulations. The TE10 mode in the WR-340 feedwaveguide is excited with a sinusoid of 2.45-GHz frequency. Theuniform cell size used is 5.83 mm and the time step was chosen tosatisfy the Courant stability condition [7]. Mur’s 2nd-order ABC[7] is used to absorb the scattered fields at the back side of the feed.The steady state was identified by looking at the power balance inthe cavity [8].

3.1. The Selective Remeshing ProcessThe mode stirrer has a periodicity of 120°. Since the computationalproblem repeats itself at every 120°, the complete effect of themode-stirrer rotation can be captured by considering one-third of asingle rotation. The 120° period is then discretized into multipleuniformly spaced mode-stirrer angular positions (such as 0°, 30°,60°, and 90°), as shown in Figure 1(b). It is generally assumed thata finer angular resolution is going to give better results [2, 3]. Thenthe FDTD simulation is started with the mode stirrer at 0° untilsteady state is reached [8]. At steady state, the power dissipation

Figure 1 Cavity details and results: (a) cavity details (the cavity has a front door with a 76.2-mm-thick material-A block as well); (b) details of the differentorientations (�LY, angle with respect to y-axis and the leaf labeled L in the x–y-plane � 0°, 30°, 60°, and 90°) of the mode stirrer considered in the study;(c) power balance in the cavity is shown with respective mode-stirrer orientations with respect to the feed waveguide. (thin black line): total; (thick blackline): normalized dissipation in material B; (thin gray line): normalized dissipation in material A; (thick gray line): normalized reflection

1180 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop

Pss inside the material is calculated from the root mean square(RMS) electric field at the center of each FDTD cell (Ecrms) as in[6]:

PSS � 2�f�0���Ecrms�2, (1)

where �� is the lossy part of the dielectric constant of the material,�0 is the permittivity of free space, and f is the excitation fre-quency. This power field is then saved for future processing.

The next step is to rotate the underlying mode-stirrer geometryabout the Z-axis. This is done by reassigning the dielectric prop-erties of the mesh containing the mode stirrer from the 0° positionto the 30° position. Remeshing is done firstly by finding thecoordinates of each Yee’s cell [7] in the new mode-stirrer positionusing the coordinate transformation matrix for rotation about theZ-axis, given by

x� � x cos��� � y sin���y� � x sin��� � y cos���

z� � z�, (2)

where � is the angle of rotation. Then the material informationstored in the mesh at location ( x, y, z) is transferred to the newlocation ( x�, y�, z�) to obtain the effect of mode-stirrer rotation.This remeshing is done “on the fly” before resuming the FDTDcalculation. When the steady state corresponding to the 30° posi-tion of the mode stirrer is reached, the power-dissipation patterninside the material is saved. The process is repeated for theremaining 60° and 90° positions of the mode stirrer and thecorresponding power-dissipation fields are saved.

Once the power-dissipation fields corresponding to all themode-stirrer positions are obtained, the time-averaged power dis-sipation inside the material in the presence of the rotating modestirrer, PMSdiss, is calculated by a method similar to [2, 3] as

PMSdiss �1

Nss�i�1

Nss

PSSi, (3)

where Nss is the total number of angular positions considered inthe simulation and PSSi is the steady-state power dissipation cor-responding to the angular position i of the mode stirrer.

Figure 2 Power-dissipation comparison: (a) geometry of the cylinder showing the two experimentally verified slices; (b) simulated electric-field dissipationat the x–y-plane; (c) measured thermal profile at the x–y-plane; (d) simulated electric-field dissipation at the x–z-plane; (e) measured thermal profile at thex–z-plane

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 1181

3.2. Convergence HistoryFigure 1(c) shows the power balance [8] in the cavity correspond-ing to different mode stirrer angular positions. Here, a value of thetotal normalized power close to 1 corresponds to steady state [8].The entire simulation was run at as a single calculation withremeshing done at multiple instances in time corresponding to thedifferent mode stirrer locations. Using the previous solution afterremeshing seems to be more efficient than running four separatesteady-state FDTD simulations of the cavity corresponding to thefour different mode-stirrer positions. Figure 1(c) shows that �150ns (350 cycles of excitation) were needed to achieve steady statefor the initial position, whereas all the remaining ones took only�90 ns (200 cycles of excitation). Here, the gain in computationaltime comes from using the previous mode-stirrer field solution,which is a closer initial guess than a zero field. In this particularexample, the remeshing and continuation approach helps finish thesimulations with 950 cycles of excitation, which would have been1400 otherwise. This can be a significant savings for long-runningsimulations.

3.3. Experimental ProcedureTwo samples of material B were obtained and sliced in half, thuscreating a rectangular cross section. These vertically sliced cylin-ders of material B were used to obtain central heating patterns atthe x–z and x–y planes [see Fig. 2(a)]. To do the experiment,thermally sensitive (fax) paper was sandwiched between the twohalves of the cylinder and inserted into the cavity. At the end of 10min, the power was turned off and the sample and fax paper wereremoved. Two different experiments were conducted for the twosamples that were split in the vertical orientations.

3.4. Comparison of ResultsFigures 2(b)–2(e) show the qualitative comparison of the resultscorresponding to the estimated FDTD power dissipation in watts/cell, and the measured thermal profiles in degrees Celsius on thethermally sensitive (fax) paper. For careful quantitative compari-son, the gray scales all the figures would have the highest intensitypoints represented by black and the lowest intensity points repre-sented by white. Unfortunately, this is not the case with the faxpaper. The fax paper fully blackens at 70°C and is insensitive tohigher temperatures, so the resolution values of the numerical andexperimental plots are not the same. Nonetheless, in all the casesthe main features of the heating pattern have been correctly pre-dicted by the model, in particular, the overall shape of the hotspots. The lack of small cold regions in the thermal-response paperof the experiment can be attributed to thermal diffusion of theapplied power and possibly to the transient nature of the experi-ment.

4. CONCLUSION

The selective remeshing approach is a new way of applying theFDTD method to microwave heating cavities with rotating modestirrers. This approach provides a way of estimating the effects ofrotating mode stirrers on the power-dissipation patterns inside thematerial being processed. By remeshing “on the fly” and continu-ing the FDTD calculation from the previously converged solution,we have achieved significant reduction in computational time.When this new scheme is applied to a �300L, cavity the resultscompare favorably to a thermal fax paper experiment.

ACKNOWLEDGMENTS

The authors thankfully acknowledge help with the experimentfrom Elizabeth Vileno, Mark Muktoyuk, and Michael Nishimoto.

REFERENCES

1. A.C. Metaxas and R.J. Meredith, Industrial microwave heating, IEEPower engineering series, Peter Peregrinus Ltd., U.K., 1993.

2. P. Plaza-Gonzalez, J. Monzo-Cabrera, J.M. Catala-Civera, and D.Sanchez-Hernandez, Effect of mode-stirrer configurations on dielectricheating performance in multimode microwave applicators, IEEE TransMicrowave Theory Tech 53 (2005), 1699–1706.

3. P. Plaza-Gonzalez, J. Monzo-Cabrera, J.M. Catala-Civera, and D.Sanchez-Hernandez, New approach for the prediction of the electricfield distribution in multimode microwave-heating applicators withmode stirrers, IEEE Trans Magn 40 (2004), 1672–1678.

4. M. Sundberg, P.O. Risman, P.-S. Kildal, and T. Ohlsson, Analysis anddesign of industrial microwave ovens using the finite difference timedomain method, J Microwave Power Electromagn Energy 31 (1996),142–157.

5. F. Liu, I. Turner, and M. Bialkowski, A finite-difference time-domainsimulation of power density distribution in a dielectric loaded micro-wave cavity, J Microwave Power Electromagn Energy 29 (1994), 138–149.

6. L. Ma, D. Paul, N. Pothecary, C. Railton, J. Bows, L. Barratt, J. Mullin,and D. Simons, Experimental validation of a combined electromagneticand thermal FDTD model of a microwave heating process, IEEE TransMicrowave Theory Tech 43 (1995), 2565–2572.

7. A. Taflove and S.C. Hagness, Computational electrodynamics: Thefinite-difference time-domain method, 2nd ed., Artech House, Boston,2000.

8. J. George, M. Muktoyuk, and R. Bergman, Enhancements to the FDTDmethod for steady state power deposition analysis in microwave heatingcavities, Microwave Opt Technol Lett 47 (2005), 530–534.

© 2006 Wiley Periodicals, Inc.

ITERATIVE-SOLVERS FOR NUMERICALANALYSIS OF PHOTONIC DEVICES

Kleucio Claudio,1,3 V. F. Rodriguez-Esquerre,3 J. P. da Silva,2,3

M. F. G. Hernandez,2 and H. E. Hernandez-Figueroa3

1 Technology Faculty of Sao Paulo StateFATEC11700-100, Praia Grande, Brazil2 Division of TelecommunicationsSuperior Center of Technological Education–CESETUniversity of Campinas, UNICAMP13484-370 Limeira, Sao Paulo, Brazil3 Department of Microwaves and OpticsSchool of Electrical and Computer EngineeringUniversity of Campinas, UNICAMP13083-970 Campinas, Sao Paulo, Brazil

Received 7 December 2005

ABSTRACT: An assessment of four linear-system iterative solvers iscarried out through the simulation of a key 2D photonic crystal struc-ture. Among these solvers, the biconjugate gradient stabilized (BCGS) inconjunction with the incomplete LU (ILU) decomposition, shows the bestperformance and can be regarded as a promising candidate for large-scale computation. In addition, the iterative solver with the best perfor-mance is applied to analysis of the directional optical coupler andtransverse anisotropic waveguide. © 2006 Wiley Periodicals, Inc.Microwave Opt Technol Lett 48: 1182–1186, 2006; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21571

Key words: photonic devices; linear systems; iterative method; aniso-tropic waveguide

1182 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 6, June 2006 DOI 10.1002/mop