Visualization of Negative Refraction in Chiral Nihility Media The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Xiangxiang Cheng et al. “Visualization of Negative Refraction in Chiral Nihility Media.” Antennas and Propagation Magazine, IEEE 51.4 (2009): 79-87. © 2009, IEEE As Published http://dx.doi.org/10.1109/MAP.2009.5338687 Publisher Institute of Electrical and Electronics Engineers Version Final published version Accessed Thu May 26 19:16:15 EDT 2016 Citable Link http://hdl.handle.net/1721.1/59282 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms Visualization of Negative Refraction in Chiral Nlhlllty Media Xiangxiang Cheng 1, Hongsheng Chen 1,2, Bae-Ian Wu2, and Jin Au Konr/'* Electromagnetics Academy at Zhejiang University Zhejiang University, Hangzhou 310058, China 1The 2Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139, USA E-mail: [email protected] *Deceased Abstract Chiral nihility media are known to produce a backward wave as one of their two polarizations, yielding negative refraction at the interface of the chiral nihility medium and free space. In this paper, the analysis of chiral nihility effects is illustrated by showing the propagation of a Gaussian beam, both reflected and refracted from an air-chiral interface, and through layered chiral nihility media that are matched to free space. The critical angle for total reflection, and the index-matched total transmission in a matched chiral half-space, are demonstrated. Also demonstrated are the wave splitting, wave widening, and a wave of "standing phase" in matched chiral nihility slabs. Keywords: Chiral media; nihility; polarization; Gaussian beams; matched media; index matched; wave splitting; electromagnetic refraction; negative refraction 1. Introduction R ecently, we have seen evidence of the fast development of the study and design of materials with negative electromagnetic parameters supporting backward waves, which are represented by popular metamaterials. As a matter of fact, backward waves can exist in more general linear media, namely in bianisotropic media, such as chiral media [1, 2], and in gyrotropic media [3, 4], which belong to different subsets of magneto-electric materials. Chiral media, which were discovered in the beginning of the 19th century, have been of much scientific interest. They have many practical applications for scientists and engineers in many different fields (e.g., physics, chemistry, and biology). The well-known natural phenomenon - optical activity - related to chiral media can rotate the polarization of an incident linearly polarized wave when propagating through a chiral medium. The polarization state of the transmitted linearly polarized wave depends significantly on the constitutive parameters' values, as well as on the thickness of the chiral medium and the incident angle [5-7]. A special case of chiral media with both permittivity and permeability equal to zero is referred as a "chiral nihility" medium [8]. This is not physically realizable in electromagnetic materials [9, 10]. Hence, in [10], the author proposed a modified condition of chiral nihility, in which the permittivity and permeability tend toward but are not equal to zero, to which we will adhere in our paper. IEEE Antennasand Propagation MagaZine, Vol. 51, No.4, August 2009 Before chiral nihility, the concept of nihility, the electromagnetic nilpotent, was introduced, in which wave propagation cannot occur, and thus the directionality of the phase velocity relative to the wave vector is a non-issue [11, 12]. Clearly, nihility is unachievable, but it may be approximately simulated at a specified frequency [13, 14]. The properties of matched zero-index passive metamaterials presented in [13] demonstrated that in the steady state, the fields inside the media are static in character, but the underlying dynamics are those of propagating transverse waves. The zero index means that the index of refraction is not matched to free space although the impedance is matched. However, in chiral nihility media, both would be matched, and the directionality of the phase velocity relative to the wave vector would not be a non-issue any more, because waves propagate in chiral nihility media. In this paper, we investigate the propagation of a Gaussian beam in a two-layer (in Section 2) and a three-layer (in Section 3) setup, the second layers of which are both chiral nihility media that are matched to free space. Different polarizations of the incident waves are considered, respectively. The waves inside and outside the chiral media are solved for analytically from Maxwell's equations by matching the boundary conditions at the interfaces. It is shown that the fields in all regions of both cases can be unambiguously calculated in closed form. Some unique characteristics of the chirality effects on energy reflection and transmission - such as total reflection, transparency, and wave splitting, which are strongly dependent on the chirality value - are observed. ISSN 1045-9243/2009/$25 ©2009 IEEE 79 2. Chiral Half Space Upon the LHCP wave incidence, two reflected waves (LHCP, RHCP), and two transmitted waves (LHCP, RHCP), are expected. The total reflected and transmitted electric and magnetic fields are expressed as follows: The constitutive relations of isotropic chiral media are [5] t , = f dkzl/f (k z)[ R- (el - e2i)e -ikixx+ikzz <:IJ (1) -00 (4) where ; =;oJEoJJo . A monochromatic time-harmonic variation exp ( -iOJt) is assumed throughout this paper, but omitted. With the application of a systematic approach named the kDE system, described in [5], two characteristic waves along with their polarization states inside the isotropic chiral media are analytically solved for. These include one right-hand circularly polarized (RHCP) wave, with wavenumber fir = 1 -00 dkzl/f(kz)[R- e\i+e2 e-ikixX+ikzz 770 +R+ -eli +e2 e-ikixX+ikzz] , 770 t; = 1 dkzl/f (k z) (5) [r-(e\ - e2"i) eik;;x+ikzz -00 and the other left-hand circularly polarized (LHCP) wave, with wavenumber (6) fit They propagate at different phase velocities of OJ/ k + and Both waves possess the same impedance, which is + - 77 = 77 = OJ/ k - . r; = 77 • V-; = 1 dkZl/f(kz)[r- e\i+e2" eik;;x+ikzz -00 77 +r+ -e\i i eik;X+ikzz 77+e J. (7) in which e2' e2" , and ei are unit vectors of the kDE systems when The + sign refers to right-hand circularly polarized (RHCP) waves, and the - sign refers to left-hand circularly polarized (LHCP) waves. Consider a half-space of an isotropic chiral medium, as shown in Figure 1, with the following incident LHCP wave : the k vectors are pointing in different directions (which means different e3 directions) : z (2) where (3) is the Gaussian spectrum that carries the information about the shape of the footprint centered at x = 0, z = 0 [15]. (At x = 0 , the amplitude at =g is 1/e times that of the center .) e\ , e2 ' and e3 Izi x are the unit vectors of the kDE system, with e3 always lying in the direction of k = e3k . Here, the incident beam is cenk; =xkix + zkiz =xkocos 0i + zkosin 0i' where 0i is the k such that tered about incident angle of the central plane wave, so that we have E, u, ~o and -i •. () • e2 =xsm i - z COS()i 80 : Figure 1. LHCP wave transmission and reflection at an isotropic chiral half-space. IEEEAntennasand Propagation Magazine, Vol. 51, No.4, August 2009 -10 6 -10 3.5 5 -5 ,..< "'N a 3 ,..< "'N 2.5 a 2 1.5 2 5 3 -5 4 5 1 1 1_Qo 0 xl>.. 0.5 1_~0 10 a xI'A 10 (b) (a) -10 -5 1.5 ,..< "'N ,..< "'N 1.5 a 1 1 5 a xl>.. 10 (e) 1_~0 0.5 a xI'A 10 (d) FIgure 2. The time-averaged power density for a left-hand circularly polarized Gaussian beam incident upon an isotropic chiral nihility half-space. (a) ~o =0.25. (b) ~o =0.5 . (c) ~o =1. (d) ~o =5. The units for the time-averaged power density shown are mW 1m 2 • IEEE Antennasand Propagation Magazine, Vol. 51, No.4, August 2009 81 ~ =1x 10-5 co' P =1x 10-5 PO ' with different chirality values. The beam propagated at an incident angle of OJ = 25° , and the electric field of the central incident wave was assumed to have a magnitude of 1 V1m. The unit for the time-averaged power density shown was ei =xsinOt -zcosot · The k~ are the x components ofwavenumbers for different characteristic waves inside the chiral medium : mW/m 2 . As predicted, Figure 2c shows the case of total transmission when the refractive index of the chiral nihility 's LHCP wave is very close to -1 . Under other chirality values, there always were LHCP reflections. Total reflection happened in Figure 2a, when the chirality, ';0' was small enough to make the critical angle where the 0l± are the refracted angles for both circularly polarized waves, which can be given by (Oe n± UI . kosin OJ =arcsin --+- . (8) k- The reflection and transmission coefficients for both circularly polarized waves, R± and t", can be found by matching the boundary conditions for the tangential (.y and z) electric and magnetic fields at the x = 0 interface. When the LHCP wave impedance matching condition, 17follows : =170 ' is satisfied, the results are as (9) R- = cos OJ - cos 01cos OJ + cos 0tT" = (10) 2 cos OJ cos OJ + cos 01- (11) These equations indicate that RHCP wave will not be excited, since the incident wave is also a LHCP wave. However, there will be a reflected LHCP field, unless it happens that either e,- =OJ or leads to R" =0 and T" =1 [2]. In chiral nihility media, while the impedance-matching condition is fulfilled, the total transmission happens when ';0 =±1, which likewise makes the index match to free space. 01- =-OJ. This The time-averaged power density can be expressed as (12) where = 14.48°) of the LHCP wave smaller than the angle of inci- dence. In the case of Figure 2b, the critical angle (Oe =30°) was larger than the wave's incident angle, so the beam could be transmitted into the chiral half space, but not totally transmitted. Along with this, Figure 2d also shows that neither total reflection nor total transmission occurred as ';0 grew larger than one. Although the reflection amplitudes were very small compared to the transmission amplitudes in Figure 2b and Figure 2d, instead of directly seeking the reflected waves, we could easily still find them through their trails in the interference stripes on the air side, near the halfspace boundary. 3. Chiral Slab In this section, the propagation of a Gaussian beam through an infinite chiral slab of thickness d in free space is considered. The configuration is shown in Figure 3. For a pure circularly polarized incident wave (LHCP or RHCP) and under the impedance-matching condition , the other circularly polarized wave (RHCP or LHCP) will not present itself while all three regions are occupied by only one characteristic wave with the same polarization as the incident wave. This is similar to what we have discussed in the previous section. However, in this section we investigate the case when the Gaussian beam is the same as in the previous section, but with a linear polarization. Upon TE incidence (or perpendicular polarization [10], or y, in our case), the reflected and transmitted fields contain both TE and TM components. This is due to boundary conditions, and the fact that waves should be circularly polarized inside the chiral slab. Using the same method and expressions in the previous section, the total electric and magnetic fields upon z E, u., ~o RHCP x LHCP Equation (12) is for arbitrary cases, once all the complex fields in each region are known . Figure 2 illustrates the time-averaged power density as a function of x and z for the impedance-matched cases where 82 Figure 3. The configuration for a chiral slab of thickness d placed in free space. IEEE Antennasand Propagation Magazine, Vol. 51, No.4, August 2009 -20 r--- y -r-- ,r -- - RCP E Distribution (VIm) RCP E Distribution (VIm) RCP E Distribution (VIm) y -, -10 -20 r----r-~I"""'"'~~rw;I -20 _ - -10 - 10 ~ o y -.-- . -.--- ""'1 ~ -0.2 -OA -0.5 0 xf)... 10 0 xf)... 10 LCP E Distribution (VIm) LCP E Distribution (VIm) y 0.5 ~ 0 o xl).. 20 10 ~ 02 o o -0.2 -0.2 -OA -0.4 -20_ 0.5 -10 ~ 0 20 10 Total E Distribution (VIm) y -r"'...",..F""'..............~ -20..-0.5 -10 o ~ y -...- ..-- - --n 0.5 - 10 o ~ -0.5 -0.5 -1 20 10 (f) Total E Distribution (VIm) y xf)... 02 (e) __- . _ . - - - _ 0 OA o w»; Total E Distribution (VIm) 1_~0 OB OA -0.5 (d) -20 _ - y ~O -10 -10 LCP E Distribution (VIm) y -20 -0.5 10 (e) (b) (a) -20 o xl).. 20 -0.5 -1 1_~0 0 v I).. (g) 10 o 20 (h) -20 -1 'i f).. 10 (i) -20 1.2 -15 -15 -10 -10 ~ -5 ~ -5 o 0.4 o 5 0.2 5 o xf)... 10 o 20 (j) time-averaged power density .u =Po , ; 0 =0.5 . 10 20 (I) Figure 4. (a)-(c) The RHCP E y component; (d)-(t) the LHCP space. (a, d, g, j): &=l xlO - xf)... s, component; (g)-(i) the total Ey component; and (j)-(l) the 1(5')1 for an obliquel y incident Gaussian beam upon three different isotropic chiral slabs in free 5&0, .u =l xlO - 5 Po, ;0 =0.7 . (h , e, h, k): &=l xlO - 5&0 , .u=l xlO- 5 Po, ; 0 = 0.5 . (c, f, i, I): &= &0' incidence can be computed. The electric fields are expressed as follows: (17) In free space, (18) (13) r. = f dkzlJf(kz)[ R- (e\ -e2i )e-ikuX+ikzZ ei2k; d 00 + B- = cos ().(cos 0..I - cos ()± ) I DE -00 + T- = Et = f dkzlJf (kz )[ r: (e\ - e~i ) eiktxx+ikzz 2e i(-k. +k±)d IX x (-1 +e i2ki d DE = =kix i) 2{1+ i 2ki d i) )cos 2 ()±. (21) Examples for Figure 3 are given in Figure 4, where the incident angle of the Gaussian beam was 25° and the thickness of chiral slab was d = 6l, where l is the wavelength of the incident wave in free space. Figure 4 shows the RHCP E y component (Fig- +A+(e\ + e~+ eik; x+ikzz + B- (e\ - et- e-ik;x+ikzz (16) ures 4a to 4c); the LHCP E y component (Figures 4d to 4f); the +B+(e\ +e~i)e-ik;X+ikzz total Ey component (Figures 4g to 4i); and the time-averaged J. ei- , ei+ , eq-, and eq+ are unit vectors of the kDB systems when the k vectors are pointing to different directions in the chiral vectors such that: media, with the same el power density, 1(8)1, (Figures 4j to 41). All are shown as a function of x and z for three contradistinctive cases where 8, P, and ~o took different values, which were all impedance-matching but not index-matching situations. One column represents one set of parameter values, respectively. The first two slabs were chiral nihility, with 8 =1x 10-5 80 and P = 1x 10-5 Po. The last slab was not a chiral nihility medium, but had k: The are the x components of wavenumbers for different characteristic waves inside the chiral media: where the ()± are the angles of refraction for both circularly polarized waves in the chiral slab, stated in the same way as Equation (8). The eight amplitude coefficients for both circularly polarized waves, R±, A±, B±, and T± can be found by matching the boundary conditions for the tangential (y and z) electric and magnetic fields at the two interfaces of the slab. This generates a linearalgebraic-equation system of eight equations. Under the impedance-matched circumstance 17 =170' all the eight coefficients can be obtained as follows: 84 )COS()i COS()± + (-1 + ei2ki d j dkzlJf(kz)[ A- (e\ -e2-i )eik;X+ikzz -00 where 2 )COs ()i - in this three-layer case. In a chiral slab, E= (20) I where -00 where ktx + cos ().cos O: DE 00 (19) ' 8 = 80 and P = Po. Since a linearly polarized incident wave is not the characteristic wave of chiral media - differently from the previous section with LHCP incidence - a TE incidence is split into two refracted waves through the chiral slab. When under oblique beam incidence, at the exit of the slab, two parallel transmitting beams with opposite circular polarizations are emitted at different locations on the second interface of the chiral slab, as shown in Figure 4. This special case can be a possible application of the slab as a wave splitter. Besides a slab, if a chiral nihility lens has a curved surface, it is able to split a linearly polarized Gaussian beam into right-circularly and left-circularly polarized beams traveling in different directions [16]. In order to obtain high wave-splitting quality and efficiency of the slab, the values of the constitutive parameters of the chiral media should be carefully selected. There is a tradeoff between the distance and the transmitted power of the two circularly polarized wave beams. As in the examples of Figure 4, under the same thickness, d, of the chiral slab and the same ~o value, chirality nihility cases are better at separating a beam with different polarizations, because one of the circularly polarized waves inside the chirality nihility media is negatively refracted. Although negative refraction also occurs in chiral media that are not chiral nihility media, it requires that the chiral parameter be large enough, or that the chirality be combined with an electric plasma [17, 18]. On the IEEEAntennasand Propagation Magazine, Vol. 51, No.4, August 2009 Figure 5. The time-averaged power density for a Gaussian beam incident upon index-matched chiral nihility slabs with angles of incidence of (a) 30 0, (b) 100, and (c) 00. The units for the time-averaged power density shown are mW / m 2 • IEEE Antennas and Propagation Magazine, Vol. 51, No.4, August 2009 85 other hand, comparing the two chiral nihility examples ( e = 1 x 10-5 &0, J.1 = 1x 10-5 J.10)' we can see that the smaller chirality value (;0 = 0.5 ) results in a larger distance of the two outward beams, but results in a lower transmission power, due to the greater mismatch in refraction index, n±::: ±~o' for both of the two characteristic waves. As was the situation for Figures 2b and 2d, the reflection amplitudes were small compared with the transmission amplitudes in Figures 4j to 41. Similarly, we can still find the reflected power through the trails in the interference patterns on the air side near the half-space boundary. Figure 5 shows the situation with the chiral nihility slab having an index matched (~o = ±1) to free space. The incident beam was totally transmitted and divided into two different polarizations on the other side of the slab. Figure 5a shows the case with the incident angle equal to 30°. Near the left interface of the slab, the two polarized transmitted beams were close to each other; interference patterns of these two beams can be clearly observed. After propagating a distance along the slab, these two beams were well separated: the structures can thus be used as a beam splitter. For smaller incident angles, as shown in Figures 5b and 5c, the two beams were not well apart inside the slab. They recombined into a linear polarization that is rotated when compared to the incident polarization, which is the famous optical-activity effect. According to Equation (28) of [7], when the slab is chiral nihility, the rotation angle becomes an equation independent of e and J.1 : E)OA = 27fd~0 cos()±. The only side effect is that the recovered beam in free space is widened, as shown in Figure 5b with a 10° incidence. However, Figure 5c shows this beam widening will not happen under normal incidence, and a wave of "standing phase" [8] is produced inside the chiral nihility slab. 4. Conclusion The problem of a tapered beam, constructed by a superposition of plane waves with a Gaussian amplitude spectrum, incident upon a matched chiral half space and a matched chiral slab, has been solved. The special case where the chiral media are chiral nihility media in both situations was discussed. An illustration of the propagation of a Gaussian beam was provided, with different chirality values. It was shown that under incidences with different polarizations, chiral nihility media that are matched to free space exhibit different wave propagating characteristics, depending on the chirality values. Such behavior can be used in lots of applications, such as beam splitting, beam widening, imaging, etc. 5. Acknowledgment The authors are grateful to Prof. Joseph R. Mautz for discussions on waves in chiral media. This work was sponsored by the Chinese National Science Foundation, under grants Nos. 60801005 and 60531020; in part by the NCET-07-0750; the ZJNSF (RI080320); the PhD Programs Foundation of MEC (No. 20070335120); the ONR under Contract No. N00014-06-1-0001; and the Department of the Air Force under Air Force Contract No. FI9628-00-C-0002. 86 6. References 1. J. B. Pendry, "A Chiral Route to Negative Refraction," Science, 306, November 2004, pp. 1353-1355. 2. C. Monzon and D. W. Forester, "Negative Refraction and Focusing of Circularly Polarized Waves in Optically Active Media," Phys. Rev. Lett., 95, 123904, September 2005. 3. J. Q. Shen, "Negative Refractive Index in Gyrotropically Magnetoelectric Media," Phys. Rev. B, 73, 045113, January 2006. 4. V. M. Agranovich, Yu. N. Gartstein, and A. A. Zakhidov, "Negative Refraction in Gyrotropic Media," Phys. Rev. B, 73, 045114, January 2006. 5. J. A. Kong, Electromagnetic Waves Theory, Cambridge, MA, EMW, 2008. 6. 1. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Hi-Isotropic Media, Norwood, MA, Artech House, 1994. 7. T. M. Grzegorczyk and J. A. Kong, "Visualization of Faraday Rotation and Optical Activity at Oblique Incidence," IEEE Antennas and Propagation Magazine, 47, 5, October 2005, pp. 23-33. 8. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, "Waves and Energy in Chiral Nihility," J. 01 Electromagn. Waves and Appl., 17, 5,2003, pp. 695-706. 9. C.-W. Qiu, H.-Y. Yao, L.-W. Li, S. Zouhdi, and T.-S. Yeo, "Routes to Left-Handed Materials by Magnetoelectric Couplings," Phys. Rev. B, 75, 245214, June 2007. 10. C.-W. Qiu, N. Burokur, S. Zouhd, and L.-W. Li, "Chiral Nihility Effects on Energy Flow in Chiral Materials," J. Opt. Soc. Am. A,25, 1, January 2008, pp. 55.. 63. 11. A. Lakhtakia, "An Electromagnetic Trinity Form 'Negative Permittivity' and 'Negative Permeability'," International Journal ofInfrared and Millimeter Waves, 23, 6, June 2002, pp. 813-818. 12. A. Lakhtakia, "On Perfect Lenses and Nihility," International Journal ofInfrared and Millimeter Waves, 23, 3, March 2002, pp. 339-343. 13. R. W. Ziolkowski, "Propagation in and Scattering from a Matched Metamaterial Having a Zero Index of Refraction," Phys. Rev. E, 70, 046608, October 2004. 14. A. Lakhtakia and J. B. Geddes ill, "Scattering by a Nihility Cylinder," AEU Int. J. Electron. Commun., 60, 11,2006, pp. 1-3. 15. J. A. Kong, B.-I. Wu, and Y. Zhang, "Lateral Displacement of a Gaussian Beam Reflected from a Grounded Slab with Negative Permittivity and Permeability," Appl. Phys. Lett., 80, 12, March 2002,pp.2084-2086. 16. S. Tariq, S. F. Mahmoud, and M. S. Laghari, "Microwave Gaussian Beam Splitting with a Variable Split Angle by Using a Chiral Lens," Radio Science, 34, 1, January-February 1999, pp. 918. IEEE Antennasand Propagation MagaZine, Vol. 51, No.4, August 2009 17. X. Cheng, H. Chen, L. Ran, B.-I. Wu, T. M. Grzegorczyk, and J. A. Kong, "Negative Refraction and Cross Polarization Effects in Metamaterial Realized with Bianisotropic S-Ring Resonator," Phys. Rev. B, 76, 024402, July 2007. 18. C. Zhang and T. J. Cui, "Negative Reflections of Electromagnetic Waves in a Strong Chiral Medium," Appl. Phys. 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