Total reflection and transmission by epsilon-near-zero metamaterials with defectsYadong Xu and Huanyang Chen Citation: Applied Physics Letters 98, 113501 (2011); doi: 10.1063/1.3565172 View online: http://dx.doi.org/10.1063/1.3565172 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/98/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Acoustic total transmission and total reflection in zero-index metamaterials with defects Appl. Phys. Lett. 102, 174104 (2013); 10.1063/1.4803919 Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zerometamaterials embedded with dielectric defects J. Appl. Phys. 113, 084908 (2013); 10.1063/1.4794011 Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials Appl. Phys. Lett. 101, 241101 (2012); 10.1063/1.4770374 Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials Appl. Phys. Lett. 100, 221903 (2012); 10.1063/1.4723844 Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control Appl. Phys. Lett. 97, 131107 (2010); 10.1063/1.3469925

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Total reflection and transmission by epsilon-near-zero metamaterials withdefects

Yadong Xu and Huanyang Chena�

School of Physical Science and Technology, Soochow University, Suzhou 215006, China

�Received 23 December 2010; accepted 20 February 2011; published online 14 March 2011�

In this work, we investigate wave transmission through an epsilon-near-zero metamaterialwaveguide embedded with defects. We show that by adjusting the geometric sizes and materialproperties of the defects, total reflection, and even transmission can be obtained, despite theimpedance mismatch of epsilon-near-zero material with free space. Our work can greatly simplifythe design of zero-index material waveguide applications by removing the dependence onpermeability. © 2011 American Institute of Physics. �doi:10.1063/1.3565172�

Recently, the manmade artificial materials denotedmetamaterials1,2 have drawn extensive attentions due to theirnumerous innovative applications.3–5 Among them aredouble negative materials,1,2,6 single negative materials,7

epsilon-near-zero �ENZ� matematerials8 and matched imped-ance zero-index materials �MIZIMs�,9 etc. Zero-index mate-rials �ZIM, including both ENZ and MIZIM�, as another typeof metamaterials, have been explored both theoretically andexperimentally to show a lot of intriguing properties.8–17 Forexample, Enoch et al.8 showed that a ZIM can enhance thedirective emission for an embedded source; Ziolkowski9

studied the possibility of designing a MIZIM; Li et al.10

proved that there is a zero-n gap inside the zero �volume�averaged refractive index material, which is distinct from theBragg gaps; Silveirinha and Engheta11–13 demonstrated anENZ medium that can “squeeze” the electromagnetic �EM�waves in a narrow waveguide. Such a tunneling effect waslater demonstrated in microwave experiments.15,16 Recently,Hao et al.18 showed that a total reflection or transmission canbe obtained by introducing perfect electric conductor �PEC��or perfect magnetic conductor, PMC� defects inside the ZIMin a two-dimensional �2D� waveguide structure. Nguyen etal.19 found that similar effects can happen when dielectricdefects are introduced into the MIZIM, which suggests anactive control transmission and reflection by incorporatingtunable refractive index materials. However, it is much morechallenging to fabricate the MIZIM than the ENZ mediumbecause it is very difficult to engineer effective permittivityand permeability to be zero at the same time. In this letter,we will revisit the problem wave transmission in a similarwaveguide structure with defects but replacing the MIZIMwith the ENZ medium. Analytic expressions will be derivedto see how total reflection and transmission can be achievedby adjusting the geometric sizes and material parameters ofthe embedded defects. Finite element numerical simulationswill also be carried out to prove our theory.

In the beginning, we would like to consider a 2D wave-guide structure, which consists of four regions as shown inFig. 1. Region 0 and 3 are free spaces separated by a ZIM�region 1� with effective permittivity �1 and permeability �1.N cylindrical defects �region 2� are embedded inside region1, whose effective permittivities and permeabilities are �2jand �2j, respectively �for the jth cylinder�. For simplicity, we

focus on the transverse magnetic �TM� polarization �the

magnetic field H� is polarized in z direction�. The walls of thewaveguide are set to be PECs �Similar results will be ob-tained for transverse electric mode where the walls of thewaveguide are PMCs�.

Suppose that a TM mode with H� inc= zH0jei�k0x−�t� is in-

cident from left to right inside the above waveguide, wherek0�=� /c� is the wave vector and � is the angular frequency.We will omit the time variation e−i�t throughout the follow-ing for convenience. The EM wave in each region followsthe Ampére–Maxwell equation:

E� n = −1

i��0�n� � H� n, �1�

where the integer n signifies each region and �n is the rela-tive permittivity of each region. The EM field in region 0 canbe written as,

H� 0 = zH0j�eik0x + Re−ik0x� , �2�

and

E� 0 =k0

��0yH0j�eik0x − Re−ik0x� , �3�

where R is the reflection coefficient. Likewise in region 3,we have,

H� 3 = zJH0jeik0�x−d�, �4�

a�Electronic mail: chy@suda.edu.cn.

FIG. 1. �Color online� The computation domain of the 2D waveguide struc-ture with a ZIM �region 1�. Region 0 and 3 are vacuum. Region 2 �thecylinders� are the embedded defects. The parallel black lines are PEC wallsof waveguide. A TM mode is incident from left to right inside thewaveguide.

APPLIED PHYSICS LETTERS 98, 113501 �2011�

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E� 3 = −1

i��0� � H� 3 =

k0

��0yJH0je

ik0�x−d�, �5�

where J is the transmission coefficient. For region 1, as its�1�0, in order to have a finite value of the electric field,19

��H� 1 must vanish, i.e., H� 1= zH1 with a constant value H1.The boundary condition at the interface of region 0 and 1gives that H0j +RH0j =H1, while another one at interface ofregion 1 and 3 is JH0j =H1. Therefore we obtain a simplerelationship between the two transmission and reflection co-efficients, J=1+R. For region 2, the magnetic field insideeach cylindrical defect obeys the Helmholtz equation,

�2H� 2 + k02�2j�2jH� 2 = 0. �6�

We will ignore the magnetic coupling between the defects asin Ref. 19. As H1 is a constant, Dirichlet boundary conditionsshould be applied at the surface of each defect and lead to themagnetic field distribution as,19

H� 2 = zH1�j=1

NJ0�k2jrj�J0�k2jRj�

, �7�

where J0 is the zero-order Bessel function of the first kind;k2j =k0

��2j�2j is the wave vector in each cylindrical defect;Rj is the radius of each cylinder; rj is relative coordinate asthat in Ref. 19. The electric field inside the defects can alsobe obtained as,19

E� 2 = iH1�j=1

NJ1�k2jrj�J0�k2jRj�

��2j

�2j� j , �8�

where J1 is the first-order Bessel function of the first kind

and � j is the azimuthal unit vector for the jth cylindricaldefect. Using Maxwell–Faraday equation,

E� dl� = − �B�

�tdS� , �9�

we can find out the transmission coefficient as,

J =1

1 −ik0�1�S − Sd�

2h−

i�

h� j=1

N �RjJ1�k2jRj�J0�k2jRj�

���2j

�2j

,

�10�

where S=d�h is total area of the region 1 and 2, Sd=� j=1

N �Rj2 is total sum of the areas of N cylindrical defects.

Let us now discuss the above transmission coefficientexpression. �a�, if �1 vanishes, the ZIM becomes a MIZIM,Eq. �10� will go back to the transmission coefficient formulain Ref. 19 �its Eq. �8��. �b�, if any of the defects is PMC��2j =−��, J=0, we can obtain similar results in Ref. 18. �c�,if �1 is finite value and no defect exists in the region 1 �i.e.,Sd=0�, the third term of the denominator in Eq. �10� willdisappear, thereby the transmission coefficient J will reduceto,

J =1

1 −ik0�1d

2

, �11�

which is in accordance with the results in Ref. 14.

According to Eq. �10�, total reflection or transmissioncan also be achieved by embedding proper defects inside theZIM. For example, if J0�k2jRj� is equal to zero, no matterwhat value of the permeability of ZIM is, there will be a totalreflection. Here we choose �1=1, such that ZIM is of a non-magnetic response, i.e., the ENZ medium which could beeasily designed and fabricated. In order to verify our analy-sis, numerical simulations are performed by using finite ele-ment solver COMSOL MULTIPHYSICS. We set d=32 mm andh=30 mm. The frequency of the incoming TM wave is 10GHz. For simplicity, here we only consider one cylindricaldefect with radius R=8 mm. The permittivity and perme-ability of the defect are chosen to be �2=2.06 and �2=1 tosatisfy J0�k2jRj�, which denotes a dielectric material.

Figure 2�a� shows the magnetic field distribution when aTM plane wave incident from the left impinges into a MI-ZIM with �1=0.01 and �1=0.01. The incoming TM wavetransmits through the structure completely. If the permeabil-ity of the MIZIM is replaced by �1=1, i.e., an ENZ medium,then only part of the incoming EM wave can transmit, asshown in Fig. 2�b�. Now, if we embed a dielectric defectdescribed above with �2=2.06 and �2=1 inside the ENZmedium, the structure will totally block the wave, as shown

FIG. 2. �Color online� �a� The magnetic distribution of a MIZIM with �1

=0.01 and �1=0.01. �b� The magnetic distribution of an ENZ medium with�1=0.01 and �1=1. �c� The magnetic distribution of an ENZ medium witha dielectric cylindrical defect with �2=2.06 and �2=1. �d� The magneticdistribution of an ENZ medium with a dielectric cylindrical defect with �2

=2.405 and �2=1.

113501-2 Y. Xu and H. Chen Appl. Phys. Lett. 98, 113501 �2011�

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in Fig. 2�c�. Likewise, the total transmission can be achievedfor the same structure by modifying the material parametersof the cylindrical defects. The total transmission happenswhen J is equal to 1. By substituting it into Eq. �10�, we findwe could simply replace the above dielectric defect with an-other one with �2=2.405 and �2=1 to fulfill the requirement.Figure 2�d� shows that such a structure has a total transmis-sion.

As the above two dielectric defects have very close per-mittivities, we can consider using tunable refractive indexmaterials �such as liquid crystals19� to control the wave trans-mission. In Fig. 3, we plot the relationship between the trans-mission efficiency and the permittivity of the dielectric de-fect �the solid curve, �2 ranges from 1 to 4�. The solid curveshows that there is a total reflection at �2=2.06 and a totaltransmission at �2=2.405, as are already shown in Figs. 2�c�and 2�d�, respectively. During the numerical simulations, wehave used an ENZ medium with �1=0.01 and �1=1. Such anapproximation is very accurate, as is also shown in Fig. 3�see the circular data, which are very close to the results fromEq. �10��. In a real ENZ medium �either designed frommetamaterials or using the plasmonic materials, such as met-als�, there should be some absorption. We consider twocases, one is an ENZ medium with �1=0.01+0.01� i, theother is with �1=0.01+0.1� i. The numerical results areshown in Fig. 3 by triangular data and star-shaped data �to-

gether with a dotted curve�, respectively. A considerableamount of loss �such as the case of �1=0.01+0.1� i� willcompromise both the total reflection and the total transmis-sion effect. However, we can still use such a structure tocontrol the wave transmission. The structure might be ap-plied in on-chip applications such as switches and opticalmodulators.

In conclusion, we have found that a total reflection ortransmission could be achieved by introducing suitable de-fects into an ENZ medium, instead of a MIZIM. Numericalsimulations confirm our theory. Compared to the MIZIM, theENZ medium is much easier to make with metamaterials orplasmonic materials, therefore gives possibilities to a fea-sible proof-of-principle experiment in future. Such a struc-ture suggests promising on-chip applications.

We thank Dr. Yun Lai and Dr. Xiaoping Hong for theirhelpful discussions. This work was supported by the Na-tional Natural Science Foundation of China under Grant No.11004147 and the Natural Science Foundation of JiangsuProvince under Grant No. BK2010211.

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FIG. 3. �Color online� The relationship between the transmission efficiencyand the permittivity of the dielectric defect. The solid curve denotes theresults obtained directly from Eq. �10�. The circular data is obtained fromnumerical simulations with an ENZ medium with �1=0.01. The triangulardata is obtained from numerical simulations with an ENZ medium with �1

=0.01+0.01� i. The star-shaped data �together with the dotted curve� isfrom numerical simulations with an ENZ medium with �1=0.01+0.1� i.

113501-3 Y. Xu and H. Chen Appl. Phys. Lett. 98, 113501 �2011�

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