IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 3, MARCH 2005 915 Electromagnetic Waves Over Half-Space Metamaterials of Arbitrary Permittivity and Permeability Akira Ishimaru, Life Fellow, IEEE, John Rhodes Thomas, and Sermsak Jaruwatanadilok Abstract—Most electromagnetic problems deal with media with unit permeability. However, recent interest in metamaterials ne- cessitated studies of wave characteristics in media with arbitrary permittivity and permeability whose real parts can be positive or negative. This paper presents analysis of wave characteristics on semiinfinite metamaterials. Waves are excited by electric or mag- netic line sources, and the problem is separated into the p (TM) and the s (TE) polarization, showing symmetries. The Fourier spectra of the reflection and transmission coefficients are examined and the poles, branch points, and zeros are shown in the real -real diagram. We clarify the location of poles in proper and improper Riemann Surfaces, and the excitation of forward and backward Fig. 1. Medium 1 is ordinary material, and both and " are real and positive. surface waves, forward and backward Lateral waves, and Zenneck Medium 2 is metamaterial with complex and " . Y ", and n are normalized waves, and the relations between Brewster’s angle and Sommerfeld with respect to medium 1. Line source is at x aHand z a h. poles. We include the behaviors of the backward surface waves and the temporal backward Lateral waves. imaginary parts should always be negative in repre- Index Terms—Brewster’s angle, lateral waves, metamaterials, sentation for passive materials. This paper presents a study of negative index, permeability, permittivity, surface waves, Zenneck waves. wave excitation in semiinfinite metamaterials. This problem for ordinary materials with has been studied extensively including the Sommerfeld poles, Zenneck waves, Brewster’s I. INTRODUCTION angle, surface waves, and lateral waves. Now, since the perme- N recent years, there has been an increasing interest in de- ability is no longer limited to one, many new waves and new I velopment of new materials whose characteristics may not phenomena emerge. be found in nature [1]–[7]. Examples are metamaterials, in par- We first present a comprehensive analysis of all CW wave ticular negative index materials (NIM), chiral media, and com- types. New wave types such as backward surface waves and posite materials. A broad range of applications has been sug- backward lateral waves are discussed and the relationship be- gested including artificial dielectrics, lenses, absorbers, antenna tween Brewster’s angle and Zenneck wave is clarified. We in- structures, optical and microwave components, sensors, and fre- clude some discussions on pulse and wave packet propagation quency selective surfaces. and time-dependent lateral waves. Our analysis is limited to In the development of these materials, there are several im- 2-D problems with isotropic metamaterials. In general, however, portant questions. We need to know how the waves behave in metamaterials are anisotropic and highly dispersive, requiring metamaterials, what characteristics may be useful for practical further investigations [5]–[9]. applications, how to construct such metamaterials, and what new applications can be identified to utilize new wave charac- II. LINE SOURCE OVER A SEMI-INFINITE METAMATERIAL teristics. This paper deals with the first of the above questions. Other questions have been considered and reported in special is- We consider an electric or a magnetic line source located over sues and other publications [6]–[9]. In conventional electromag- a metamaterial as shown in Fig. 1. netics, the permeability is one For the p-polarization (TM), we have except magnetic materials. In metamaterials, however, the real parts of and can range from to , though the (1) Manuscript received March 4, 2004; revised September 8, 2004. This work was supported by the National Science Foundation under Grant ESC-9908849. where is the magnetic line current and we use the normal- The authors are with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). ization is free space wavenumber, and Digital Object Identifier 10.1109/TAP.2004.842572 are relative permeability and permittivity . 0018-926X/$20.00 © 2005 IEEE 916 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 3, MARCH 2005 Fig. 2. (a) Regions of b I (white), I b b H (gray), and ` H (dark). (b) Different regions in " - plane. Using the boundary condition that and are continuous at , we get the well-known Fourier rep- resentation of the incident , the reflected and the transmitted fields [11]. (2) where is the wave impedance. First we note that for the s-polarization (TE), we have the same equation for with the replacement of by . The boundary condition at is the continuity of and . Therefore the field for the s-polarization is identical to the field for the p-polarization with the interchange of and in both media. III. SURFACE WAVES AND LATERAL WAVES As preliminary to this analysis, we need to pay close atten- tion to the square root of various quantities. For passive media, we require, with time dependence and where denotes the imaginary part of. The refrac- a@k A ` H a@k A b H tive index needs to be chosen such that Fig. 3. Regions for (proper) and (improper) in the " - plane. a" a (a) 10. (b) 2. (c) 1. (d) 0.5. (e) 0.1. (3) Let us examine the reflection coefficient From (3), it follows that the characteristic impedance and admittance must have positive real parts, where (4) . Next, needs to be chosen carefully. From the branch The zero and pole are given by points at in the complex plane, we draw the branch cuts along . Then, in the top surface of plane, (5) and the wave attenuates as . Thus this is called the proper Riemann surface. Even though the branch Solving for , we get the pole at . cuts can be drawn in other ways, the above choice is most common [10]. In the improper Riemann surfaces, one or both become positive [11]. (6) ISHIMARU et al.: ELECTROMAGNETIC WAVES OVER HALF-SPACE METAMATERIALS 917 Fig. 4. (a) Forward surface waves and backward surface waves . (b) Forward lateral waves v and backward lateral waves v waves. NIM is in the third quadrant. To verify whether the pole is in the proper Riemann surface, we first obtain (7) The square root is taken such that . We then calculate using (5) (8) If , the pole is in the proper Riemann surface (I), and if , the pole is in an improper Riemann surface (II). See [11]. Note that in (6) corresponds to both zero and pole in (4). Fig. 5. Zenneck wave pole at o and Brewster’s zero at x. Riemann surface I: Brewster’s angle is when the reflection coefficient is zero and, a@k A ` HY a@k A ` H. II: a@k A b HY a@k A ` H. III: a@k A ` therefore, the classification as pole or zero is based on (5). We HY a@k A b H.IV:a@k A b HY a@k A b H. will discuss this further later. Let us now examine the location of the pole in the -plane. For real and is pure real or pure imaginary so the pole lies on the real or imaginary axis in the -plane. Fig. 2 then shows the regions in the - plane where different wave types can exist. If , forward or backward surface waves can exist. If , there may be Zenneck waves, and if , the poles are in the imaginary axis and the wave may be exponen- Fig. 6. Propagation constant k at (a) Brewster’s zero. (b) Zenneck wave tially decaying along the surface. pole. However, this is not sufficient to describe the wave types. We and backward surface waves exist in the two regions shown in also need to examine whether these poles are in the proper or Fig. 4(a).We also note the forward and the backward improper Riemann surfaces by examining whether is lateral waves can exist in region and , negative or positive with . However, this depends respectively, [see Fig. 4(b)]. greatly upon the ratio of the imaginary parts of and .We write IV. BREWSTER’S ANGLE AND ZENNECK WAV E It has been known that Brewster’s angle and Zenneck wave (9) are closely related. However their relationship is often not We show in Fig. 3, five cases where the ratio clearly explained. We first note that at Brewster’s angle, the . reflection coefficient is zero while at the Zenneck wave pole In Fig. 2, there are four regions where and surface (sometimes called the Sommerfeld pole), the reflection coeffi- waves can exist. However, as shown in Fig. 3, the two regions cient is infinite. Brewster’s angle is normally defined for plane (a and D) with on the right side of Fig. 2 wave incidence on lossless dielectric material, but here, we are in the improper Riemann surface. Hence, the surface wave generalized it to include a lossy medium and complex angle, can exist only in the region where the poles are in the and call it Brewster’s zero. This terminology was included in proper Riemann surface. In fact, we will show that the forward (5). 918 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 3, MARCH 2005 Fig. 7. Complex -plane @ a k ak A plots of pole and branch cut to n for (a) case p X 4 aPXR HXIjY" aP HXHIjY a@4Aaa@"AaIH. (b) Fig. 8. Complex -plane @ a k ak A plots of pole and branch cut to n for case p X 4 aPXR HXHIjY" aP HXIjY a@4Aaa@"AaHXI. (c) case @Y "A cases in A, C, A ,B of Fig.