Blanchard et al. Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2117
Response of dispersive cylindrical cloaksto a nonmonochromatic plane wave
Cédric Blanchard,1,2,* Bae-Ian Wu,2 Jorge Andrés Portí,1 Hongsheng Chen,2,3 Baile Zhang,2 Juan Antonio Morente,1
and Alfonso Salinas1
1Department of Applied Physics, University of Granada, 18071 Granada, Spain2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3The Electromagnetic Academy at Zhejiang University, Zhejiang University, 310027 Hangzhou, China*Corresponding author: [email protected]
Received August 25, 2009; revised August 25, 2009; accepted September 2, 2009;posted September 21, 2009 (Doc. ID 108771); published October 20, 2009
. INTRODUCTIONhe possibility of excluding all electromagnetic (EM)elds from a certain region without perturbing the vicin-
ty has been extensively studied since the pioneer workroposed by Pendry et al. [1]. Based on a coordinate trans-ormation, the cloak presented in the related original pa-er must disperse and can be wholly efficient only at aingle frequency. For this reason, most of the works deal-ng with cloaking have long been focused on monochro-
atic EM waves [2,3], which elude the dispersion effect.owever, the authors recently looked into more physicalispersive cloaks [4]. In particular, it has been theoreti-ally shown that the frequency center of a quasi-onochromatic wave is blueshifted in the forward direc-
ion after passing through a spherical cloak [5].The central aim of this paper is to employ a full-wave
umerical analysis of the response of a cylindrical cloako a quasi-monochromatic EM wave in order to illustrateow some phenomena emerge from the dispersive naturef the device. The simulation will be carried out using theransmission line modeling (TLM) method. The TLMethod can model the EM field by filling the space with
odes made of intersecting transmission lines [6,7]; as aime domain method, TLM can well describe the physicalspect of the cloaking process [8].The first part will deal with TLM of cloaking struc-
ures. In its original form, TLM exploits the well-known–C distributed network [9] representation of homoge-eous dielectrics and magnetic materials in which theuantities L and C represent positive equivalent perme-bility � and permittivity �, respectively. Simply inter-hanging the position of L and C leads to an equivalentispersive material that assumes negative values for both
and � [10]. Therefore, we will focus on the dispersive na-ure of TLM mesh for metamaterials.
Second, it will be shown that the frequency center of auasi-monochromatic wave is blueshifted in the forwardirection for the cylindrical cloak after passing throughhe cloaking structure. It will be demonstrated that therequency shift distribution depends on the observationngle; in particular a redshift occurs in some directions.Leonhardt first stressed that the cloaking device causestime delay [11], which has been deeply studied after-ard by using ray-tracing simulations [12]. Such a geo-etric optics description is valuably interesting, but the
act that it constitutes an approximation (reflection is in-vitably ignored, for instance) requires deeper analyses.e propose to confirm this physical aspect of the cloaking
ynamics in a full-wave simulation. Furthermore, timeelays associated with cloaks based on nonlinear coordi-ate transformations will also be simulated.
. TRANSMISSION LINE MODELINGETHOD
he coordinate transformation,
r� = f��r� = ��r − R1�/�R2 − R1��1/�R2, �1�
hich can compress the space from 0�r��R2 to the con-entric annular region R1�r�R2, yields the well-knownormulas [2],
�r = �r =r − R1
r,
009 Optical Society of America
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2118 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Blanchard et al.
�� = �� =r
r − R1,
�z = �z = � R2
R2 − R1�2r − R1
r, �2�
f �=1. These r-dependent quantities lead to the idealloak at a single functional angular frequency, which weill refer to as �0 in this paper.Dealing with quasi-monochromatic waves is delicate
ince they do not differ much from perfectly monochro-atic waves; consequently, numerical errors in the mod-
ling have to be minimized. In this sense, Cartesian coor-inates, which are usually employed [13,14], lead totaircase approximations when one wants to model theurved shape of the layered cloak. To avoid this undesir-ble phenomenon, we will use TLM cylindrical nodeshose shape assumes the geometry of the cloaking shell
15,16]. Therefore, with such nodes, it is not necessary topproximate the curved geometry with many Cartesianells anymore. Furthermore, since �� and �� tend to in-nity at the inner boundary, numerical simulations use todopt approximations arising from the truncation of thenner layer given that they cannot deal with infinite val-es. With the TLM, dielectric and magnetic constantsending to infinity do not have any detrimental conse-uence since it can be proven that the limits of the TLMcattering matrix elements still remain finite in this case17]. That is why this extreme value will be used in theodeling, which will contribute to the improvement of the
ccuracy of our simulation.As an illustration of the reachable accuracy, let us con-
ider a 2 GHz EM wave incoming on a cloaked perfectlectric conductor (PEC) cylinder. The cloaking shell, withn inner radius of R1=0.1 m and an outer radius of R20.2 m, is made up of 200 layers. The polarization is
ransverse magnetic (TM), which means that the mag-etic field is oriented along the axis of the cylinder (z di-ection). The computed magnetic field distribution is dis-layed in Fig. 1(a). Note that we have used the Huygensurface technique that consists of dividing the mesh into aotal-field inner region and a scattered-field outer region;bsorbing boundary conditions at the outer boundary ofhe mesh can thus be employed, while both the total andcattered fields can be visualized in the same picture. It islain from Fig. 1(a) that the cloaking effect can bechieved with a great precision by using TLM, with cylin-rical nodes, proposed here. For instance, the scatterings shown to be almost zero in region 4. Quantitatively, thear-field scattering radiations for the cloaked cylinder andor the bare cylinder are also shown in Fig. 1(b). It turnsut that the scattering for the cloaked cylinder has beeneduced by 40 dB in the forward and backward directions,nd even by more along other directions, compared withhe simple cylinder scattering.
The above example involved a monochromatic wave,ut the aim of this paper is to deal with nonmonochro-atic waves. Therefore, an important issue is in regard to
he dispersion of the cloaking material in the numericalodeling. According to Eq. (2), the cylindrical cloaking
hell is made up of an anisotropic material that involves
ifferent categories of EM constants: the azimuth compo-ent is always greater than 1, the radial component is al-ays less than unity, while the axial component is eitherreater than 1 or less than 1 on both sides of the limitalue r=R2
2 / �2R2−R1�. Given that the interconnection be-ween all the TLM nodes constitutes a usual distributed–C network, TLM cannot process the exotic values of �nd � in its original form. The problem can be fixed byubstituting the L–C network by a dual one, i.e., the po-itions of inductors and capacitors are simply inter-hanged [10,13]. Such a left handed transmission line sys-em is dispersive, which is in agreement with theecessary dispersive nature of metamaterials. Thus, theLM mesh, which is nothing more than the numerical in-arnation of these networks, has also a dispersive naturef � and � are less than unity. Consequently, it is not nec-ssary to artificially make � and � dispersive; both al-eady are given that they are implemented through theispersive mesh. An important remaining worry is whats the exact frequency behavior of the EM constants in theylindrical mesh? We can calculate that the dispersionenerated by the mesh is
ig. 1. (Color online) (a) Magnetic field ��10−3� mapping in theicinity of a cloaked PEC cylinder at 2 GHz. Four regions are ap-arent: 1, cylinder; 2, cloaking shell; 3, free space with total field;, scattered-field region where incident field is subtracted. (b)ar-field pattern comparison between the cloaked and the bareylinders.
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Blanchard et al. Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2119
�z��� = Az�r� −�0
2
�2 �Az�r� − �z��0��,
�r��� = Ar�r� −�0
2
�2 �Ar�r� − �r��0��, �3�
here Az and Ar are two functions of r that also depend onhe parameters of the TLM. They are given by the formu-as
Az�r� =2�t�zZ0
r�r���0, Ar�r� =
�t�rY0
r���z�0, �4�
n which �t is the time step of the numerical simulation;r, ��, �z represent the size of the nodes; and Z0 and0=1/Z0 represent the impedance and the admittance,espectively, of the connecting transmission lines consti-uting the node. Let us consider a dispersive mediumhose permittivity and permeability, represented by ���,
ollow the Drude model ���=1−�p2 /�2 [18], where �p is
he plasma frequency. From the last equation, we findhat �p
2=�02�1−��0�� so that ���=1−�0
2 /�2�1−��0��;ote that, if the dispersive medium under consideration iscylindrical cloak, ��0� is given by Eq. (2). We conclude
hat �z��� and �r��� in Eq. (3) are reminiscent of a Drudeodel [18]; the factor 1 is, however, substituted by the
unction Az or Ar. Since Az and Ar depend on parametersf the numerical simulation, we will have to ensure thatz��� and �r��� are in agreement with the causality con-ition d������ /d�1. This condition can be derived fromhe Kramers–Kronig relations, which are a direct conse-uence of the causality principle [19]. Note that Yaghjiannd Maci showed that considerations involving the EMnergy conservation are capable of providing the same in-quality, but they moreover proved that the relation������ /d���� should be employed for metamaterials20]. Although in some studies the Lorentz model haseen employed for the permeability [21], other contribu-ions dealing with left handed metamaterials [22] orloaking structures [14] do use the Drude model [18]. Inhis sense, it is worth emphasizing that the Drude model18] employed in this paper does not alter any conclusionshat we will obtain from the numerical simulation.
. FREQUENCY SHIFT FOR AUASI-MONOCHROMATIC WAVE
or a time domain method as TLM, a quasi-onochromatic EM wave can be modeled by using aodulated sinusoid in terms of time t,
h�t� = sin��t�exp�− g2�t − tm�2�, �5�
here �=2�f is the angular frequency (f is the frequency)f the incoming wave, while g and tm are two real num-ers that define the bandwidth of the signal. The Fourierransform of h is depicted in Fig. 2. Obviously, such an in-oming wave behaves as a quasi-monochromatic waveiven that its frequency domain representation is aaussian pulse centered on �. The quasi-monochromaticature of the wave is controlled by the parameter g: themaller the g constant, the narrower the pulse in the fre-
uency domain. Therefore, if g is chosen to be smallnough, any scattering due to the dispersion inherent tohe cloak is minimized around the cloaking frequency;nd, for this reason, the frequency domain representationf the wave that passes through the cloak, i.e., the trans-itted wave, should be a Gaussian too.Let us consider a PEC cylinder surrounded by a cloak-
ng shell whose functional frequency is f0=2 GHz. Thetructure is illuminated by a TM wave. To see exactly howhe cloaking effectiveness is affected by a deviation of therequency, we first compute the scattering width (SW) ofhe cloaked PEC cylinder in terms of the frequency. Theesult is depicted in Fig. 3 and compared with the SW of aimple PEC cylinder. As expected, the cloak is effectivenly for the f0 working frequency. However, the SW is ob-iously very low in a certain bandwidth centered on f0 andor which the object is still undetectable. The size of suchbandwidth depends on the sensitivity of the detector lo-
ated outside the structure. In this sense, the bandwidthhat may be considered as quasi-undetectable containshe whole frequency range of the plane wave plottedn Fig. 2. Consequently, we will use this quasi-
onochromatic wave, with �=2��2 GHz and g=5107 s−1, in the numerical determination of the fre-
uency shift that follows.
ig. 2. Frequency domain representation of the incoming wavefunction h) for �=2��2 GHz and g=5�107 s−1.
ig. 3. (Color online) SW of a PEC cylinder with/without theloaking shell around. The scattering is reduced in a narrowbandround the 2 GHz working frequency.
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2120 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Blanchard et al.
While a strictly monochromatic wave would pass theylindrical cloak without distortion, the situation mighthange slightly for a quasi-monochromatic wave even ifhe bandwidth is narrow as is the case here. First, let usxamine what happens in the forward direction. The Held is computed at a distance �0 (incoming radiationavelength) from the outer boundary of the cloak. Fur-
hermore, in order to illustrate how the response is sensi-ive to the assumed dispersion, two different sets of�z��� ,�r���� are considered. Concretely, TLM parametersre set such that Ar�r�=0.10/r and Az�r�=0.13/r in onease, and Ar�r�=0.21/r and Az�r�=0.13/r in the otherase. As an example, we plot �r�� /2�� in Fig. 4 at the vi-inity of the inner boundary of the cloak, precisely at r0.11 m, for the two dispersion profiles. Therefore, theorresponding frequency domain representation, obtainedy Fourier transform, is displayed in Fig. 5 for both rela-ions of dispersion. The frequency centers of the newulses result to be shifted toward higher frequencies; thisorresponds to a blueshift of the wavelength. Concretely,t turns out that the frequency centers of the pulses haveeen shifted by 0.45 and 0.74 MHz depending on the as-umed dispersion. This shows that the frequency shifthenomenon is a direct consequence of the dispersion ofhe optic constants, and higher slope values lead to morentense blueshift effects. To provide some insights on theelationship between the profile of the constitutive pa-ameters and the observed frequency shift, let us considerig. 6 in which �r�� /2�� has been plotted at r=0.1, 0.11,nd 0.12 m. It is plain from this figure that there exists arequency for which �r=0 for the three curves. Of course,t r=R1, the frequency that makes �r equal to zero is theorking frequency of the cloak, in agreement with Eq. (2).owever, because of the assumed dispersion, when r in-
reases �r reaches the zero value for lower frequencies.he same effect is observed for �z. These zero values of
he optic constants lead to a concentric wall for the EMave that enables it to go further toward the inneroundary of the cloak. Thus, while the higher frequencyblue light) components of the quasi-monochromaticource are capable of reaching the PEC core, the loweromponents (red light) are scattered by the wall before,
ig. 4. (Color online) Two different curves of dispersion for theonstitutive parameter �r at r=0.105 m. Curves 1 and 2 corre-pond to A �r�=0.10/r and 0.21/r, respectively.
r
.e., for r R1. As evinced in Fig. 6, the apparent radius ofhe wall is different depending on the frequency compo-ent under consideration. Accordingly, the scattered and
ncident fields should interfere differently depending onhe frequency. Moreover, each layer of the cloak is de-cribed by a different dispersion profile. Therefore, the in-eractions inside the cloak are complex and a numericalreatment may be useful. To understand how they inter-ere, the cloak is illuminated by a monochromatic wavehose frequency is slightly deviated from the cloaking
requency (2 GHz). Concretely, the frequencies of the in-oming wave are chosen to be 1.99 and 2.01 GHz; the re-ults are displayed in Figs. 7(a) and 7(b), respectively. Webserve that scattering does exist in these two cases, buthe interaction between the incident and the scatteringelds is not the same: (1) for 1.99 GHz, the total field is
ess intense than the incident field, meaning that thecattered field destructively interferes with the incidenteld; (2) for 2.01 GHz, the scattering strengthens the in-
ig. 5. (Color online) Frequency domain representation of theransmitted wave in the forward direction. The frequency centersf the pulses are blueshifted. The red dashed curve correspondso Ar�r�=0.10/r (i.e., curve 1 in Fig. 4), while the blue dottedurve corresponds to Ar�r�=0.21/r (i.e., curve 2 in Fig. 4). The fre-uency shift depends on the relation of dispersion.
ig. 6. (Color online) Dispersion of the r component of the per-ittivity at three different points inside the cloak. At r=R1, �r
akes, of course, the zero value at the working frequency (2 GHz).ut when r increases, the frequency that renders � =0 increases.
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Blanchard et al. Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2121
ident field, which leads to a more intense total field. Thiseans that, on one hand, the amount of red light tends to
iminish and, on the other hand, the amount of blue lightncreases. That is why the blueshift effect occurs overall.
We now numerically calculate the frequency shift alongther directions; the result is depicted in Fig. 8. The shift
(b)
(a)
ig. 7. (Color online) Incident, scattered, and total fields for aloak whose working frequency is 2 GHz. It is illuminated by anM plane wave with frequencies of (a) 1.99 GHz (red light) and
b) 2.01 GHz (blue light). For the red light, a destructive inter-erence is observed, while the interference is constructive for thelue light.
ig. 8. (Color online) Distribution of the shifted frequency cen-er in terms of the observation angle for a quasi-monochromaticM wave traveling from left to right.
ontinuously spans a large spectrum, going from negativeo positive values. The corresponding redshift occursround 35°. Such a distribution can be explained in termsf constructive and destructive interferences as well. Inhat sense, the ratio between the amplitudes of the totalnd incident fields, for f=1.99 and 2.01 GHz, is plotted inig. 9. In the forward direction, in agreement with Fig. 7,his ratio is greater than unity for 2.01 GHz while it isess than unity for 1.99 GHz. However, the predominancef the blue light over the red light can be inversed forther angles. In particular, for f=1.99 GHz and in theange of 25°–45°, the scattering constructively interferesith the incident wave given that we observe a greatermplitude of the total field than the amplitude of the in-ident field. This tendency is reversed for the blue light sohat, overall, a redshift of frequency is expected with aaximum at 32°, in agreement with Fig. 8. It is impor-
ant to insist that it is the specific employed dispersionhat leads to this particular angular distribution. Indeed,he dispersion profile is directly related to the apparentize of the EM wall (for f� f0) and, therefore, to the kind ofnteraction between the scattered and incident fields forach frequency component.
. TIME DELAYSecause the EM constants of the cloak are r dependent,
he wave velocity is expected to follow a certain distribu-ion: Chen and Chan showed that the velocity decreaseshen getting closer to the inner boundary of the invisible
hell for a spherical cloak [12]. What about the cylindricalloak? Let us consider a TM wave incident upon a cylin-rical cloak; the incoming magnetic and electric fields cane expressed as
Hzinc�r� = H0 expi
�
cr cos � ,
Erinc�r� = E0 expi
�
cr cos �sin �,
E�inc�r� = E0 expi
�
cr cos �cos �. �6�
rom Eq. (6), the EM field in the cloak can be derived us-ng [19]
ig. 9. (Color online) Ratio between the total and incidentelds, for f=1.99 and 2.01 GHz, in terms of the direction ofbservation.
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2122 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Blanchard et al.
Hz = Hzinc�f�r��,
Er = f��r�Erinc�f�r��,
E� =f�r�
rE�
inc�f�r��, �7�
here f�r� is the transformation function in Eq. (1). Fromq. (7), the total EM energy density can be obtained as
19]
ti(ntdtt
gn
qtTtttn
idttiufpAwfixpwrlrSftr
W̄ =1
4�0
d���ik�
d�EiEk
� + �0
d���ik�
d�HiHk
� , �8�
hile the mean value of the Poynting vector is
S̄ =1
2�Re�E � H���. �9�
rom Eqs. (8) and (9), we can get the velocity according to= S̄ /W̄ [19]; finally, the calculation gives
u = c
2R2�� r − R1
r �2
cos2 � + sin2 �
�R2 − R1��R2 sin �
R2 − R1�2d���r�
d�+ � r − R1
r
R2 cos �
R2 − R1�2d�����
d�+
d���z�
d� , �10�
here c is the speed of light in free space. For the purposef illustration, the distribution of the velocity in a cloakhat is dispersive in accord with the Drude model [18] isepicted in Fig. 10 for �=0°, 45°, and 90°. It is plain fromhis last figure that the velocity is smaller near the inneroundary of the cloak. Thus, once the EM wave hasassed through the cloak, the process of reaching differ-nt points located on a plane normal to the direction ofropagation should be completed with a time delay.Equation (10) has been derived in order to make intui-
ive the fact that time delays must exist. However, itould be incomplete to use such an approach based oneometric optics (ray-tracing analysis) to determine theagnitude of time delays. Only a full-wave analysis can
rovide rigorous information for the reasons mentioned inhe following:
(1) For a monochromatic plane wave there would beatching of the impedance at the interface of two adja-
ent layers. However, because of the dispersion, it is not
ig. 10. (Color online) Velocity distribution in a dispersive cloaklong three different directions. The optics constants �z��� and��� of the cloak follow a Drude model [18].
he case for the nonmonochromatic plane wave involvedn that study. That is why the cloaking device scatterseven if the scattering is small) if it is illuminated by aonmonochromatic plane wave. The consequence is thathe plane wave interacts with an object, the cloak, whoseimensions are similar to the wavelength. As a result theime delays are mainly due to diffraction, which cannot bereated by geometric optics and rays of light.
(2) Since the cloaking shell is anisotropic and inhomo-eneous, the direction of propagation of the EM wave isot necessarily the same as those of the ray of light.(3) In Section 3, it has been shown that the low fre-
uency components cannot reach the inner boundary ofhe cloak, while the high frequency components can.herefore a serious distortion of the signal is expected at
he vicinity of the inner boundary. A ray, which is only aheoretical object, cannot take into account such a distor-ion, while the physical nonmonochromatic wave defi-itely can.Time delays cannot be computed if the incoming wave
s monochromatic given that the cloak would induce noistortion in this case; in other words, the incident andhe transmitted signals would overlap making impossiblehe distinction between each. Instead, suppose that thencoming wave is quasi monochromatic and is modeledsing the modulated sinusoid of Eq. (5), with the cloaking
requency chosen to be f0=6 GHz. The permittivity andermeability are given by Eq. (3) with Ar�r�=0.10/r andz�r�=0.13/r. Note that these parameters are those thatere used to get the red curve 1 in Figs. 4 and 5. The Held is calculated in the forward direction at the distance0=�0 from the cloak outer boundary. In Fig. 11, we com-are the H field in the absence and in the presence of thehole structure in terms of time, normalized by the pe-
iod of the incident wave, T0=1/ f0. Obviously, if the enve-ope of the signal is considered, the presence of the cloakesults in the EM wave to reach x0 with a time delay.ince the wave frequency is high, oscillations resulting
rom the sine in Eq. (5) cannot be observed and we have,herefore, enlarged a certain portion of Fig. 11.
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Blanchard et al. Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2123
It is important to study the response of cloaking devicesased on nonlinear coordinate transformations [23], i.e.,ith values of � not equal to 1 in Eq. (1). Let �2 be thelane normal to the x axis at point x0. Let ttd be the de-iation between the required times for the EM wave toeach �2 with and without the presence of the wholetructure. Let us consider the line, contained in �2, whichoints along the y direction. Depending on the position ofhe point under consideration, ttd is expected to take dif-erent values. In the following, we propose to compute ttdnormalized by T0) in terms of the distance y from the xxis. To calculate ttd, we look at the EM power associatedith the signal. We first calculate the total power P0 of
he incident wave. Then, we admit that the required timeor the transmitted wave to reach �2 is when the power0/2 has flowed through �2. The result is depicted in Fig.2 for �=1, 0.75, and 0.5. As expected, ttd decreases wheneviating from the x axis. Furthermore, the time delay islainly sensitive to this type of transformation, with theinear one rendering the delay smaller. For completion’sake, we have tried other frequencies for the incomingave. The obtained curves turn out to be different from
hose presented in Fig. 12. The reason is that the phe-omenon mainly comes from diffraction; thus the time de-
ig. 11. (Color online) H-field plot in terms of the normalizedime. The envelope of the signal does not reach x0 at the sameime depending on whether the cloaking structure is present orot.
ig. 12. (Color online) Time delay (normalized by T0) in terms ofhe distance y from the x axis.
ay has to depend on the size of the cloak or, equivalently,n the frequency of the incident wave.
We pointed out the deficiency of the ray-tracing modelt the beginning of this section. Therefore, discrepanciesetween Eq. (10) and our numerical results are expected.or example, it is plain from Fig. 10 that the velocity is aonotonically increasing function of the radial compo-ent r. On the contrary, Fig. 12 evinces that the time de-
ay is constant in a certain range. Furthermore, if thebove parameters are modified so that Ar�r�=0.21/r andz�r�=0.13/r (as it has been done in Section 3), themount of time delay should be altered. Concretely, weave verified that the time delay is increased in the for-ard direction, which cannot be entirely explained in the
imit of geometric optics. Indeed, as pointed out in [12], aay would take nearly infinite time to propagate along theropagation axis given that the velocity approaches zerot the inner boundary of the cloak (see Fig. 10). These twoomments illustrate that the ray approach is unable toake into account all the material effects as diffraction oristortion of the wave.It is worth noting that our numerical simulations show
hat there is a slight pulse broadening. This result wasxpected given that the constitutive parameters of theloak disperse, which means that each frequency compo-ent propagates with a particular velocity. However, therequency band of the incident wave is narrow. Thereforehe constitutive parameters do not suffer an importantariation and the pulse broadening cannot be directly ob-erved in Fig. 11. As evinced in Fig. 4, using Ar�r�0.21/r and Az�r�=0.13/r increases the slope of the dis-ersion curve around the working frequency. It has beenhecked that these parameters lead to a wider pulseroadening.Finally let us emphasize that the physical origin of
oth time delays and frequency shifts is the dispersion ofhe constitutive parameters of the cloak. If the cloak wasot dispersive, neither time delays nor frequency shiftsould be observed. Nonetheless, they differ in the mecha-ism. Time delay is related to diffraction while the fre-uency shift is related to the apparent size of the concen-ric wall for each frequency component of the incidentave. In addition, the frequency shift reflects the magni-
ude of the transmission for each frequency while theime delay reflects the phase profile of the transmissionor each frequency.
. CONCLUSIONhe interest in the response of a cloaking shell to a non-onochromatic plane wave derives from the dispersiveature of the invisible device; confining the study to oneingle frequency eludes this aspect. In this sense, the full-ave simulation of a cylindrical cloak illuminated by auasi-monochromatic EM wave is able to reveal and con-rm interesting phenomena.We report in this paper the TLM of such a configuration
y using cylindrical nodes. We showed first that thisreatment provides significantly precise results. Whenassing through a cylindrical cloak, the frequency centerf a quasi-monochromatic wave is blueshifted in the for-ard direction. This phenomenon is known for a spherical
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2124 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Blanchard et al.
loak but had not yet been demonstrated for its two-imensional equivalence. In other directions, shifts withther intensities occur; we found out in particular that aedshift can take place. Another aspect is in regard toime delay. It is shown that the EM wave envelope exhib-ts a delay in the process of reaching a plane located at aonstant distance from the device. Furthermore, time de-ays, which mainly come from diffraction, strongly dependn the coordinate transformation that yields the param-ters of the cloak.
CKNOWLEDGMENTShis work has been supported in part by the “Ministerioe Educación y Ciencia” of Spain under research projectsIS2004-03273 and FIS2007-63293, cofinanced withEDER funds of the European Union (EU). H. Chen ac-nowledges the support of the National Natural Scienceoundation of China (NSFC) under grant 60801005, theoundation for the Author of National Excellent Doctoralissertation of People’s Republic of China under grant00950, the Zhejiang Provincial Natural Science Founda-ion under grant R1080320, and the Ph.D. Programsoundation of MEC under grant 200803351025.
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