circuit controlled by a pulse-waveform comparison algorithm, OpticalFiber Communications (OFC) Conference, San Jose, 1994 (PaperTuN4).

3. S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. Starodubov, A. Willner,and J. Feinberg, Adjustable compensation of polarization mode disper-sion using a high-birefringence nonlinearly chirped fiber Bragg grating,IEEE Photon Technol Lett 11 (1999), 1277–1279.

4. I. Abe, H.J. Kalinowski, R.N. Nogueira, J.L. Pinto, and O. Frazao,Production and characterisation of Bragg gratings written in high-birefringence fibre optics, IEE Proc Circuits Devices Syst 150 (2003),495–500.

5. A. Othonos and K. Kalli, Fiber Bragg gratings–Fundamentals andapplications in telecommunications and sensing, Artech House, Nor-wood, MA, 1999.

© 2006 Wiley Periodicals, Inc.

HIGH-POWER EMISSION, NARROW-BEAM SCANNING, AND SUB-WAVELENGTH IMAGING USINGCONDUCTING GRATINGS AND LEFT-HANDED MATERIALS

Zhuo Li, Tie Jun Cui, and Jian Feng ZhangDepartment of Radio EngineeringCenter for Computational Electromagnetics and the State KeyLaboratory of Millimeter WavesSoutheast UniversityNanjing 210096, People’s Republic of China

Received 19 April 2006

ABSTRACT: Many amazing properties of a novel electromagnetic(EM) structure, consisting a conducting grating backed with a left-handed material (LHM) slab, are studied. High-directivity, high-poweremission and narrow-beam scanning can be easily realized when thisstructure acts as an antenna. Simultaneously, if it works as an imagingsystem, extremely high resolution can also be obtained with the uniqueEM behaviors of the LHM. Proper usage of surface waves on aperturesopened in the conducting screen makes this structure possess such at-tractive capabilities. In order to investigate such phenomena, a moregeneral EM model is first developed through the equivalence principle,and then the method of moments is implemented to obtain the interestingresults. All these amazing properties are clearly observed through thenumerical results. © 2006 Wiley Periodicals, Inc. Microwave OptTechnol Lett 48: 2359–2365, 2006; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.21944

Key words: left-handed materials; imaging; high-power emission; nar-row-beam scanning

1. INTRODUCTION

Transmission of electromagnetic (EM) waves through apertures inan infinitely thin, perfectly conducting (PEC) plane has alreadybeen well studied [1]. Based on the diffraction theory, if only oneelectrically small hole exists, the transmission power will be verysmall since it is mostly reflected by the PEC plane, and the emittedbeam is very broad because the EM waves are diffracted to alldirections uniformly. Recently, some approaches have been devel-oped to enhance the transmission power with a narrow beam [2–5].In [2–4], surface resonance formed by the periodic corrugations isthe substantial reason to cause such magic phenomenon. Differentfrom [2–4], LHM is introduced to achieve even better performancein [5].

Although the theory foundation of the LHM has already beenestablished by Veselago in 1968 [6], in which the unusual physicalproperties of this novel material are anticipated, such as negativeindex, high resolution imaging, etc., very few researches are per-formed in this field because this material has not been found innature even today. In recent years, there has been some degree ofhype associated with the same capability realized by many kinds ofartificial material [7–10]. Investigations of sub-wavelength imag-ing have received increased interests and attention, with the sig-nificant progress made in manufacturing more practical structures,to realize the LHM properties. Although the perfect lens proposedby Pendry [11] has proved to be unphysical [12, 13], sub-wave-length imaging can still be achieved when the LHM is slightlylossy [14–16].

Different from [5], an infinite long line source parallel to y-axisand polarized in x-direction is considered, i.e., only TM modesexist. The high-directivity and high-power emission are first in-vestigated, and then methods to realize narrow-beam scanning arediscussed. Finally, the demonstration of the sub-wavelength im-aging property of this EM structure is performed. In the followingsection, equivalence principle is implemented to obtain the inte-gro-differential equation which is used in method of moments(MoM) to solve such problems numerically. Some interestingresults are then given in Section 3 to verify our analysis.

2. PRINCIPLES AND FORMULATIONS

A general two-dimensional (2D) problem is considered as shownin Figure 1, where a conducting grating is inserted in a stratifiedmedium. Obviously, the whole space is divided into two halfspaces by the conducting grating. For the convenience of laterdiscussion, the sign l/r is used as the first subscript of all symbolsin Figure 1 to indicate that the parameter belongs to the left/righthalf space. The permittivity and permeability of each region R(l/r)i

are denoted by �(l/r)i and �(l/r)i, respectively. In order to excite TMmodes only, an infinite long line source Ji parallel to y-axis andpolarized in �-direction is considered, where �� � xx � zz. Theconducting grating is assumed to lie in the x-y plane, and aperturesare infinite in the y direction. The total electric and magnetic fieldsin each region R(l/r)i are denoted as E(l/r)i and H(l/r)i, respectively.

Based on the equivalence principle [17], this problem can bedecomposed into two sub problems: the left-region equivalenceproblem and the right-region equivalence one. For the left-regionequivalence problem, unknown equivalent magnetic current den-

Figure 1 A conducting grating immersed in multiple-layered media,where an arbitrary electric source is located in Region RlN

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sities 2M on the apertures are introduced to maintain the validityof such equivalence, and the factor 2 in 2M arises from theimplementation of the image theory. Similarly, unknown equiva-lent magnetic current density � 2M are introduced in the right-region equivalence problem, in which the sign � in � 2M mustbe added to maintain the continuity of the tangential electric fieldon such apertures. For more detail discussions, one can refer to [5].

After the original problem is decomposed, the total magneticfield in the left region 1, Rl1, can be expressed as

Hl1 � H1li � H1

le, (1)

in which H1li is the incident magnetic field, and H1

le is the scatteredmagnetic field due to the equivalent magnetic current density 2M.We remark that H1

li is actually excited by the original source Ji andits image since the image theory is applied.

Similarly, the total magnetic field in the right region 1, Rrl, canbe expressed as

Hr1 � H1re, (2)

where H1re is the scattered magnetic field due to the equivalent

magnetic current densities � 2M. In the original problem, thetangential magnetic fields should be continuous across the aper-tures in the conducting screen; hence we have

z � Hl1 � z � Hr1 (3)

on the apertures. By substituting Eqs. (1) and (2) into (3), weobtain the following integro-differential equation:

z � �H1li � H1

le��z�0 � z � H1re�z�0 . (4)

Since the continuity of the electric fields on apertures has alreadybeen satisfied as mentioned above, only Eq. (4) is needed todetermine the unknown magnetic current densitiesM. Owing topaper length consideration, explicit expressions for H1

li, H1le, and

H1re are omitted, which can be derived from the multilayered

medium theory developed in Chapter 2 of Ref. 18. In this paper,MoM is applied to solve the integro-differential Eq. (4) numeri-cally. Once the unknown magnetic current densities M are ob-tained, any interested field variables can be easily computed.

3. SIMULATIONS AND DISCUSSIONS

In this section, high-directivity, high-power emission, narrow-beam scanning, and sub-wavelength imaging of this EM model arestudied. As mentioned in [5], in order to realize such properties,some layers of the stratified medium must be LHM. In the follow-ing simulations, a relative simple situation is considered, i.e., onlyone LHM slab is used. The relative permittivity and permeabilityof the LHM slab are both set to be �1 � i10�5, and thecoordinate of each point is given in the form of �x,z� with the unitof meters. The source term in Figure 1 can be written as Ji��s�� xIl��x � xs���z � zs�, in which �xs,zs� is its location, Il� 10�3 A � m denotes the current moment, and the operatingfrequency is 1 GHz.

3.1. High-Directivity and High-Power EmissionAs we all know, if the conducting screen is located in free spaceor backed with a conventional right-handed material (RHM) di-electric slab, the equivalent magnetic current densities on theapertures will be small, this in turn makes the transmission powervery small.

Based on the unusual physical properties of LHM, strongsurface waves are anticipated to be existing on the two boundariesof the LHM slab. To verify this anticipation, we first assume onlyone LHM slab is embedded in the free space. The distribution ofthe x component of the electric field is shown in Figure 2, in whichthe source is located at (0, �0.15) and the boundaries of the LHMslab are located at z � 0.0 and z � 0.3 respectively. From Figure2, very strong surface waves on the slab boundaries are clearlyobserved. As mentioned in [5], if these strong surface waves areproperly used, high-directivity and high-power transmission can beobtained simultaneously. To make use of these strong surfacewaves, a conducting grating residing on the right boundary of theLHM slab is then introduced, and apertures are opened at peaks ofthe surface waves according to Figure 2. In this case, the equiva-lent current densities M on these apertures will be very large dueto the strong surface waves, which in turn make the transmissionpower extremely larger than that when the LHM slab is removed.To demonstrate the above analysis, a single-aperture case is firstconsidered. According to Figure 2, the center of the aperture ischosen to be at (0, 0.3) where the amplitude of the surface wavesachieves the maximum value and the aperture width is set to be0.0670. The far-field pattern of the electric field in this case isillustrated in Figure 3, in which the source location and the LHMboundaries are both the same as that in Figure 2. In order to clarifythe effect of the LHM slab, far-field pattern of the electric field inthe case where the LHM slab is removed and the source is movedto (0, 0.15) is also computed, as shown in Figure 3. Since only oneelectrically small aperture is considered, far-field pattern of theelectric field looks more like a circle, which can be clearly ob-served in Figure 3. However, by use of the LHM slab, �Ex� is about6.6 times larger than that without the slab, which makes theemitted power density 43.56 times larger.

Although the emitted power has been enhanced greatly in thesingle-aperture case, the directivity is not good. In order to obtaina narrow beam, an eight-apertures case is taken into consideration,where centers of the eight apertures are chosen to be located at(�0.56, 0.3), (�0.315, 0.3), (�0.14, 0.3), (�0.07, 0.3), (0.07, 0.3),(0.14, 0.3), (0.315, 0.3), and (�0.56, 0.3) respectively. These

Figure 2 Distribution of electric fields (amplitudes) inside and outsidethe LHM slab excited by the 2D electric dipole, which correspond to theTM mode. The conducting grating does not exist here. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com]

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points are just the peak positions of the surface waves, which willgenerate a tapered distribution of the equivalent magnetic currents.In the eight-apertures case, �Ex� in the near field region is computedto show the expected tapered distribution, as illustrated in Figure4, and the far-field pattern of the electric field is shown in Figure5. Again, we remove the LHM slab and move the source to (0,0.15), then the far-field pattern of the electric field is calculated asa comparison. From Figure 5, it is evident that the directivity of theemitted beam is very good and the power density is nearly 100times larger when the LHM slab is introduced.

From the numerical simulations, we find that the proper usageof the strong surface waves is the key to realize the high-directivityand high-power emission.

3.2. Narrow-Beam ScanningTill now, the high-directivity and high-power emission has beenwell studied, which can be realized by a conducting grating backedwith a LHM slab. In this section, how to use such structure to

Figure 3 The far-field patterns of Ex components emitted by the 2Delectric dipole through a single-aperture grating backed with/without theLHM slab. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

Figure 4 The near-field distribution of Ex component in a region close tothe LHM slab emitted by the equivalent magnetic current densities throughthe eight-aperture grating. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com]

Figure 5 The far-field pattern of Ex component emitted by the 2Delectric dipole through the eight-aperture grating backed with/without theLHM slab. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

Figure 6 Narrow-beam scanning though the conducting grating andLHM slab by moving the 2D electric dipole up and down in which Xs

represents the x coordinate of the source. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com]

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realize high-power, narrow-beam scanning is investigated. Basedon the antenna array theory, beam scanning can be realized bycontrolling the relative phase of each array element. From theequivalence principle, apertures in the PEC plane act as an arrayantenna. Hence, beam scanning can be expectantly obtained bychanging the relative phase of the equivalent currents on suchapertures. One way to change the relative phase of the equivalentcurrents is to change the location of the source. For example, thesource can be moved up and down or back and forth.

To verify our analysis, the far-field patterns of the electric fieldare computed when the source is moved up and down, as illus-trated in Figure 6. In Figure 6, the blue line, red line and green linerepresent cases when the source is located at (�0.05, �0.15), (0,

�0.15), and (0.05, �0.15) respectively. It is clear that the high-power, narrow beam scanning is realized within a certain anglescope (from �14° to 14°). Obviously, if the single source isreplaced by a phased array, even better performance is expected.However, such a proposal may be expensive because multipledipoles will be required.

3.3 Sub-Wavelength Imaging EffectsBased on the analysis by Pendry and others [11–16], nearly allpropagating components and most of evanescent components ofthe fields radiated by the source can be recovered at the exteriorimage point formed by the small lossy LHM slab. In this section,the sub-wavelength imaging property of the conducting gratingbacked with a LHM slab is studied and some interesting phenom-ena are found. As we know, the LHM slab acts as a super lens if

Figure 7 A single-aperture conducting grating resided on the left bound-ary of the LHM slab, where the aperture is centered at (0, 0), and the 2Delectric dipole is located at the point (0,�0.15)

Figure 8 The magnetic fields 239 Hy 239 along the line x � 0 from z �0 to 9 m when the aperture size varies from 0.2 to 1.00 with the step 0.20.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

Figure 9 The magnetic fields 239 Hy 239 along the line x � 0 from z �0 to 0.9 m when the aperture size varies from 7.0 to 10.00 with the step1.00. [Color figure can be viewed in the online issue, which is availableat www.interscience.wiley.com]

Figure 10 The magnetic fields 239 Hy 239 along the line x � 0 from z �0.45 m to z � 0.9 m when the aperture size varies from 0.2 to 1.00 withthe step 0.20. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

2362 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 11, November 2006 DOI 10.1002/mop

the conducting grating does not exist, and so the amplitude of themagnetic field �Hy� in this ideal case can be used as a reference.Therefore, in any other case, the closer to that in the ideal case theamplitude of the magnetic field �Hy� is, the well the imagingproperty is expected. Owing to the paper length consideration,only the single-aperture case is considered in the following dis-cussions.

First, the conducting grating is assumed to reside on the leftboundary of the LHM slab and the source is located at (0, �0.15)as shown in Figure 7. The thickness of the LHM slab is chosen tobe 0.3 m. Thus, there will be an image point at (0, 0.45) when onlythe LHM slab exists. Obviously, if the single-aperture conductinggrating is introduced, the location of this image point will bechanged, which also means �Hy� at this point will be different fromthat in the ideal case. In order to watch the aperture effect on �Hy�around the expected image point more clearly, �Hy� is only evalu-ated on the z-axis (x � 0) from now. We anticipate that the largerthe aperture size is, the better the imaging property will be, whichis verified by our numerical simulations as shown in Figures 8 and9. The aperture size is varied from 0.20 to 1.00 by step of 0.10

and from 7.00 to 10.00 by step of 1.00 in Figures 8 and 9respectively, and the computation region is from z � 0.0m to z� 0.9m in both Figures. Clearly, Hv is very different whenaperture size changes from 0.20 to 1.00, especially at z� 0.3m, which is the right boundary of the LHM slab. Thisdramatic difference of �Hy� at z � 0.3m can be explained as thatthe very strong surface waves on the right boundary of the LHMslab are more sensitive to the aperture size. When the aperture sizeis beyond 6.00, however, �Hy� is nearly unchanged as illustrated inFigure 9. Simultaneously, Figures 10 and 11 are also given as theamplified images of Figures 8 and 9 in order to watch such changesmore clearly around the image point. From these two figures, it isevident that �Hy� at the exterior image point changes slightly in allcases, which indicates that �Hy� at the image point is less sensitiveto the aperture size. Base on the above analysis, sub-wavelengthproperty of the signal-aperture conducting grating is expected andhas been demonstrated in Figure 12, where �Hy� is evaluated at theimage point with different aperture size. As shown in Figure 12,

even in the case of small aperture (e.g., 1.60), the imagingproperty is very good.

In the above discussion, only the influence of the aperture sizeon the imaging property is studied. In addition to this kind ofinfluence, the influence of the distance between the source and theLHM slab is also studied in the following part. The consideredstructure is almost the same as that in Figure 7 except that theconducting grating is moved to the right boundary of the LHM slabas illustrated in Figure 13. To investigate the effect of the distancebetween the source and the LHM slab, the source position isassumed only to be changed along the z-axis. Such distance is firstfixed to be a certain value, and then �Hy� at the exterior image pointis evaluated with different aperture size as shown in Figure 14, in

Figure 11 The magnetic fields 239 Hy 239 along the line x � 0 from z �0.45 m to 0.9 m when the aperture size varies from 7.0 to 10.00 with thestep 1.00. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

Figure 12 The magnetic fields 239 Hy 239 at the exterior image pointwhen the aperture size varies from 0.1 to 100. [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com]

Figure 13 A single-aperture conducting grating resided on the rightboundary of the LHM slab, where the aperture is centered at (0, 0.3), andthe 2D electric dipole is located in the left region of the slab

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 11, November 2006 2363

which the source location is changed from zs � � 0.13m tozs � � 0.04m by step of 0.010 and the aperture size is changedfrom 0.20 to 10.00. We remark that �Hy� is evaluated at differentpoints for different distances since the image point will move asthe source moves. As illustrated in Figure 14, the sub-wavelengthimaging can be realized even when the aperture size is small andwhen the source is not close to the LHM slab (e.g., zs �� 0.13m). In order to obtain the same property of imaging,however, the aperture size needs to be larger when the source isclose to the LHM boundary (e.g., zs � � 0.04m). This inter-esting phenomenon can be easily explained as the farer the sourceis from the LHM slab, the closer the interior image point is to theconducting grating, and the smaller the aperture needs to be tocatch sufficient information carried by the incident waves. Suchconclusion will be reversed when the source moves towards theLHM slab. From the above analysis, we find that besides high-directivity and high-power emission, the sub-wavelength imagingcan also be realized by the conducting grating backed with a LHMslab, which may be more useful in practice.

4. CONCLUSIONS

In this work, some attractive EM properties of a conducting gratingbacked with a LHM slab are studied. When this structure works asan antenna, high-directivity, high-power emission, and narrow-beam scanning can be simultaneously achieved by opening aper-tures in appropriate positions in the conducting plane. As shown in

our numerical results, the efficient usage of the strong surfacewaves on such apertures is the key to obtain such very goodperformance. In addition, when the same structure acts as animaging system, sub-wavelength imaging can also be realized evenin the small aperture case, which may be more useful in practice.

ACKNOWLEDGMENT

This work was supported in part by the National Science Founda-tion of China for Distinguished Young Scholars under Grant No.60225001, in part by the National Basic Research Program (973)of China under Grant No. 2004CB719802, in part by the NationalScience Foundation of China under Grant No. 60496317, and inpart by the National Doctoral Foundation of China under GrantNo. 20040286010.

REFERENCES

1. J.D. Jackson, Classical electrodynamics, 3rd ed., Wiley, New York,1998.

2. T. Thio, H.J. Lezec, and T.W. Ebbessen, Strongly enhanced opticaltransmission through subwavelength holes in metal film, Phys B 279(2000), 90–93.

3. H.J. Lezec, A. Degiron, E. Devaux, R.A. Linke, L. Martin-Moreno,F.J. Garcia-Vidal, and T.W. Ebbesen, Beaming light from a subwave-length aperture, Science 297 (2002), 820.

4. L. Martin-Moreno, F.J. Garcia-Vidal, H.J. Lezec, A. Degiron, andT.W. Ebbesen, Theory of highly directional emission from a single

Figure 14 The magnetic field 239 Hy 239 at the exterior image point when the aperture size varies from 0.1 to 100, where the 2D dipole moves fromzs � �0.13 to �0.04 m. Here, 239 zs 239 represents the distance between the dipole to the left boundary of the LHM slab. The reference value (blue line)indicates the result when the conducting grating does not exist. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]

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subwavelength aperture surrounded by surface corrugations, Phys RevLett 90 (2003), 167401.

5. J.F. Zhang and T.J. Cui, High-directivity and high-power emission ofa line source through conducting gratings backed with a slab ofleft-handed material, Phys Rev B 72 (2005), 125123.

6. V.G.Veselago, The electrodynamics of substances with simulta-neously negative values of � and �, Sov Phys Usp 10 (1968), 509.

7. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, Extremely lowfrequency plasmons in metallic mesostructures, Phys Rev Lett 76(1996), 4773.

8. R.A. Shelby, D.R. Smith, and S. Schultz, Experimental verification ofa negative index of refraction, Science 292 (2001), 77.

9. A. Grbic and G.V. Eleftheriades, Overcoming the diffraction limit witha planar left-handed transmission-line lens, Phys Rev Lett 92 (2004),117403.

10. C. Caloz and T. Itoh, Electromagnetic metamaterials: Transmissionline theory and microwave applications, Wiley, New York, 2004.

11. J.B. Pendry, Negative refraction makes a perfect lens, Phys Rev Lett85 (2000), 3966–3969.

12. G.W.’t Hooft, Comment on negative refraction makes a perfect lens,Phys Rev Lett 87 (2001), 249701.

13. N. Garcia and M. Nieto-Vesperinas, Left-handed materials do notmake a perfect lens, Phys Rev Lett 88 (2002), 207403.

14. S.A. Ramakrishna and J.B. Pendry, The asymmetric lossy near-perfectlens, J Mod Opt 49 (2002), 1747.

15. T.J. Cui, Z.C. Hao, X.X. Yin, W. Hong, and J.A. Kong, Study of lossyeffects on the propagation of propagating and evanescent waves inleft-handed materials, Phys Lett A 323 (2004), 484.

16. Q. Cheng, T.J. Cui, and W.B. Lu, Lossy and retardation effects on thelocalization of EM waves using a left-handed medium slab, Phys LettA 336 (2005), 235.

17. R.F. Harrington, Field computation by moment methods, Macmillan,New York, 1968.

18. W.C. Chew, Wave and fields in inhomogeneous media, 2nd ed., IEEEPress, New York, 1995.

© 2006 Wiley Periodicals, Inc.

CORRIGENDUM: IDEAL AND REAL PBGMULTILAYERS OF NEGATIVE ANDPOSITIVE PHASE VELOCITYMATERIALS IN A PARALLEL-PLATEWAVEGUIDE

Ricardo A. Depine,1 Marıa L. Martınez Ricci,1 andAkhlesh Lakhtakia2

1 Grupo de Electromagnetismo Aplicado, Departamento de Fısica,Facultad de Ciencias Exactas y Naturales, Unıversidad de BuenosAires, Ciudad Universitaria, Pabellon 1, 1428 Buenos Aires, Argentina2 CATMAS—Computational and Theoretical Materials SciencesGroup, Department of Engineering Science and Mechanics, 212Earth and Engineering Sciences Building, Pennsylvania StateUniversity, University Park, PA 16802-6812

Received 28 March 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21817

It was brought to our attention that there were several errors in the arti-cle originally published. We apologize for the inconvenience.

Please note that the online version has been corrected.

ABSTRACT: Originally published in Microwave Opt Technol Lett 48:(2006); 1945–1953. Comparison of guided wave propagation (a) in a par-allel-plate waveguide with perfectly conducting walls and filled with eitherreal or ideal photonic bad gap (PBG) multilayers whose unit cell comprisesa positive phase velocity and a negative phase velocity layers, and (b) inthe stratification direction as well as obliquely in the same PBG multilayersbut of infinite transverse extent assists in the explanation of several band-gap features and reflectance peaks, particularly those associated with zero

average refractive index, computed for (a). © 2006 Wiley Periodicals, Inc.Microwave Opt Technol Lett 48: 2365–2372, 2006; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22022

Key words: Bragg condition; Fabry–Perot condition; multilayer; nega-tive phase velocity; photonic band gap; parallel-plate waveguide

1. INTRODUCTION

One-dimensional photonic band gap (PBG) materials have had arich history in electromagnetics and optics for several decades [1,2], and the recent fabrication of negatively refracting materials[3–5] have renewed interest in them [6]. The phase velocity andthe time-averaged Poynting vector are oppositely directed in iso-tropic, negatively refracting materials. Although many names havebeen given to these materials, it is best to call them as negativephase velocity (NPV) materials [7–9], their conventional counter-parts being positive phase velocity (PPV) materials.

Electromagnetic propagation along the direction of stratification inmultilayers containing NPV materials has been investigated by a fewresearchers [10–14]. Gerardin and Lakhtakia [10] showed that theBragg regime would shift to shorter wavelengths, if one of the twoconstituent materials in the unit cell of a conventional PBG multilayerwere to be replaced by its NPV analog. The same authors [11] studiedthe spectral responses of fractal multilayers containing NPV materi-als. Photon tunneling in the presence of a NPV layer was demon-strated by Zhang and Fu [12], and unusual narrow transmission bandsin the Bragg regime of PBG multilayers consisting of alternating PPVand NPV layers were reported [13, 14]. When the constitutive prop-erties of the PPV and the NPV constituents of a unit cell in such aPBG multilayer are such that the average refractive index of the unitcell is null valued, the structure naturally becomes invariant upon achange of scale length and insensitive to disorder [13]. If, in addition,the two layers are impedance matched, each unit cell becomes equiv-alent to nihility [15–17], a medium underlying the concept of Pend-ry’s perfect lens [18, 19].

The NPV containing PBG multilayers considered thus far [10–14]employ layers of infinite transverse extent, to our knowledge. Thisassumption is satisfactory in the optical regime, wherein the trans-verse dimensions of typical devices are considerably bigger than theoptical wavelengths. However, this assumption cannot be practicallyscaled down to the microwave and the milimeter wave regimes,wherein waveguides must be used. The difference is substantial formultilayers containing NPV materials, since most of the availableNPV materials operate at relatively low frequencies, typically in theGHz range, and only recently has the terahertz gap been bridged.

Our purpose here is to examine the electromagnetic responsesof PBG multilayers consisting of alternate PPV and NPV layersinserted longitudinally inside a parallel plate waveguide with per-fectly conducting walls. The structure is reminiscent of the Kro-nig–Penney morphology that has been examined by Gomez et al.[20] for PBG multilayers containing nonmagnetic PPV materials.We must also distinguish between ideal and real PBG multilayers:whereas the former comprise an infinity of unit cells, the lattercontain a finite number of unit cells [21].

The plan of the article is as follows. Section 2 summarizes thederivation of the dispersion equations for transverse electric (TE)and transverse magnetic (TM) modes inside ideal PBG multilayersinserted in parallel plate waveguides, as well as the method used toobtain the transmittance of a real PBG multilayer inserted in aparallel plate waveguide. Section 3 is devoted to numerical results.Concluding remarks are provided in Section 4. An exp(�it) timedependence is implicit throughout the article, with the angularfrequency, t the time, and i � �� 1.

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