- Email: [email protected]
Total reflection of electromagnetic waves propagating from an isotropic medium to an indefinite metamaterial
Optics Communications 274 (2007) 248–253 www.elsevier.com/locate/optcom
Total reflection of electromagnetic waves propagating from an isotropic medium to an indefinite metamaterial Yuanjiang Xiang, Xiaoyu Dai, Shuangchun Wen
*
School of Computer and Communication, Hunan University, Changsha 410082, China Received 16 July 2006; received in revised form 20 November 2006; accepted 3 February 2007
Abstract The existence conditions for total reflection and the corresponding critical angle at the interface separating an isotropic medium and an indefinite metamaterial for TE- and TM-polarized electromagnetic waves are obtained. For different kinds of indefinite metamaterial, there appear different total reflection phenomena. Particularly, the anomalous total reflection in which the incident angle is smaller than the critical angle and the Brewster’s angle can be smaller than the critical angle can occur for anti-cutoff medium. Furthermore, the omnidirectional total reflection exists for the always cutoff medium and anti-cutoff medium. The total reflection depends on the thickness of indefinite metamaterial when the indefinite metamaterial is finite, and the photon tunneling phenomenon can occur when the thickness of indefinite metamaterial is smaller than wavelength. Ó 2007 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Bs; 42.25.Gy Keywords: Indefinite medium; Negative index material; Total reflection; Photon tunneling
1. Introduction The negative index material (NIM) with negative permittivity and negative permeability simultaneously, first introduced by Veselago [1], has recently received much attention in the literatures owing to its very unusual properties [2–5], such as negative refraction index [5] and wave vector antiparallel to the Poynting vector [1]. NIMs were realized primarily in the microwave range. Nowadays, NIMs in the near IR and optical range have also been experimentally demonstrated [6,7]. Several applications have been envisioned for these materials: perfect lens [3], compact-cavity resonators [8], phase shifters [9], and efficient waveguide [10,11]. Recently, the characteristics of electromagnetic waves propagation in anisotropic NIM, even an indefinite metamaterial [12] in which not all the principal elements of the permeability and permittivity ten*
Corresponding author. Tel./fax: +86 731 8823474. E-mail address: [email protected] (S. Wen).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.013
sors have the same sign also have been discussed extensively. Schurig and Smith have outlined a potentially useful application of the indefinite metamaterial as a spatial filtering [13]. Hu and Chui have mainly discussed under what conditions anomalous reflection or refraction can occur at the interface when propagating waves pass from one isotropic regular medium into another uniaxially anisotropic NIM medium [14]. Furthermore, Shen and Wang predicted the existence of a total transmission phenomenon at the interface separating a regular material and an indefinite metamaterial under suitable conditions [15]. In classical electromagnetic field theory, when light travels from the optically dense material to the optically rarer material, the total reflection will occur if the incident angle is larger than the critical angle [16]. This phenomenon has been found applications in attenuated total reflection spectroscopy [17], scanning photon tunneling microscopy [18], and microscale thermophotovoltaic devices [19]. In this paper, we investigate the problem of total reflection of
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
electromagnetic waves at the interface formed by an isotropic regular material and an indefinite metamaterial. We present a detailed analysis on the existence conditions for total reflection for all the possible physical parameters. This total reflection phenomenon is significantly different from that for isotropic materials. This paper is organized as following. We first investigate the propagation characteristic at the interface between an isotropic medium and an indefinite metamaterial in Section 2. Then we discuss the existence conditions for the total reflection and the critical angle when the light goes from the isotropic medium to the indefinite metamaterial in Section 3. In Section 4, we analyze the condition for total reflection when the indefinite metamaterial is finite, and present a numerical result. Finally, in Section 5, we give our conclusion. 2. Propagation characteristics of electromagnetic wave at the interface with indefinite metamaterial The geometry of the considered problem is shown in Fig. 1, where the plane wave of angular frequency x is incident from the medium 1 at an incident angle h into the medium 2 (indefinite metamaterial). The medium 1 is characterized by the permittivity e1 and the permeability l1, and the indefinite metamaterial is characterized by the permittivity e, the permeability l. The interface of the two media is parallel to the xy plane, and the normal direction is the z-axis. To simplify the proceeding analysis, we assume the anisotropic permittivity and permeability tensors can be simultaneously diagonalizable, then e and l tensors can be denoted as the following forms: 0 1 0 1 ex 0 0 lx 0 0 B C B C e ¼ @ 0 ey 0 A; l ¼ @ 0 ly 0 A: ð1Þ 0 0 ez 0 0 lz In the subsequent analysis, we assume that the indefinite metamaterial is optically uniaxial, and the optical axis (zaxis) is normal to the interface, then ex ¼ ey 6¼ ez and lx ¼ ly 6¼ lz . Without loss of generality, we assume that the wave vector locates at the xz plane, and the incident electric field is E ¼ ^y E0 eiðkx xþk1z zxtÞ , where kx and k1z are x and z components of the incident wave vector. We first discuss the TE
249
modes, and the TM modes can be treated similarly. For the TE waves, the dispersion relation in the isotropic regular medium is k 2x þ k 21z ¼ e1 l1 ðx2 =c2 Þ;
ð2Þ
and for the indefinite metamaterial, it is k 2x k2 x2 þ 2z ¼ 2 : ey lz ey lx c
ð3Þ
Thus the normal wave vector component in the indefinite metamaterial satisfies, k 22z ¼ ðx2 =c2 Þ½ey lx ðlx =lz Þe1 l1 sin2 h:
ð4Þ
In the absence of loss, the sign of k 22z can be used to distinguish the nature of the plane-wave solutions. k 22z > 0 corresponds to real valued k2z and propagating solutions. k 22z < 0 corresponds to imaginary valued k2z and exponentially growing or decaying (evanescent) solutions. It is noted that the sign of the wave vector k2z must ensure the Poynting vector inside the indefinite metamaterial to point away from the interface between the incident and indefinite metamaterial, and according to reference [20], k2z and lx must keep the same sign. The reflection coefficients and transmission coefficients can be found by considering the boundary conditions for the electric field and magnetic field at the interface lx k 1z l1 k 2z ; lx k 1z þ l1 k 2z 2lx k 1z ¼ : lx k 1z þ l1 k 2z
rTE ¼
ð5Þ
tTE
ð6Þ
Similarly, we also easily obtain the reflection coefficients and transmission coefficients for TM waves: ex k 1z e1 k 02z ; ex k 1z þ e1 k 02z 2ex k 1z ¼ ; ex k 1z þ e1 k 02z
rTM ¼
ð7Þ
tTM
ð8Þ
where 2 2 2 k 02 2z ¼ ðx =c Þ½ex ly ðex =ez Þe1 l1 sin h:
ð9Þ
If the optical axis of the indefinite metamaterial is parallel to the interface between the two media, we can also choose the z-axis to be normal to the interface and the x-axis to be along the optical axis, then ex 6¼ ey ¼ ez and lx 6¼ ly ¼ lz . Finally the reflection coefficient and transmission coefficient can be obtained similarly. It is easily found from Eqs. (5)–(9) that the total reflection occurs when 2
RTE ¼ jrTE j ¼ 1
ð10Þ
for TE waves and when RTM ¼ jrTM j2 ¼ 1 Fig. 1. Geometrical structure of the problems. The optical axis of the indefinite metamaterial is z-axis.
ð11Þ
for TM waves. In the conventional medium, at the interface formed by two regular media, when the incident angle
250
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
larger than the critical angle, the z-component of the wave vectors of the refracted waves becomes imaginary and the total reflection occurs. In contrast, if the interface is associated with an indefinite metamaterial, the situation should be quite different because the permeability or the permittivity of the two media across the interface also could be different. In next section, we detailedly analyze the conditions of the total reflection occurrence and the conditions of critical angle. 3. The condition for occurrence of total reflection According to Eqs. (10) and (11), it can be found that there may have a value of the wave vector k2z, which can make the reflection coefficient RTE or RTM equal to be 1. After some algebra derivations, k2z must satisfy the following condition: k 22z ¼ 0: Hence, the critical angle can be obtained, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ey lx hc ¼ sin1 : ðlx =lz Þe1 l1
ð12Þ
For the cutoff medium, the total reflection can occur under some conditions, according to the inequality (14), the condition is h > hc :
ð18Þ
This condition is the same as the condition for total reflection phenomenon in the conventional medium, i.e., that the incident angle must be larger than the critical angle, and it is the normal total reflection. We have given an example in Fig. 2. Furthermore, we can find that the Brewster’s angle is existent under some conditions even for the TE waves, and that the Brewster’s angle is smaller than the critical angle. For the anti-cutoff medium, as the physical parameters ey lx ; lx =lz are simultaneously negative, thus from the inequality (14) we can find that if the following condition is satisfied: h < hc
ð19Þ
then k2z will be imaginary and the incident waves can be totally reflected. This condition is obviously different from
ð13Þ
The corresponding value of kx for which k 22z ¼ 0 is the cutoff wave vector separating propagating from evanescent solutions, we denote it as kc. From Eq. (4), this value is pffiffiffiffiffiffiffiffi k c ¼ k 0 ey l z . Then if k 22z < 0, the total reflections are always satisfied, viz., k2z becomes an imaginary number, so ey lx ðlx =lz Þe1 l1 sin2 h < 0:
ð14Þ
Next we discuss the existence possibility of total reflection for different physical parameters. The first restriction of critical angle comes from the span of sin2 hc , thus the following inequality set must be satisfied: 2
0 < sin hc < 1:
ð15Þ
The above inequality is, in fact, equivalent to the following inequality: 0 < ey lz < e1 l1 :
Fig. 2. Variation of reflectance with incident angle for an indefinite medium (cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2 (solid line), ey ¼ 2; lx ¼ 1; lz ¼ 1 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line).
ð16Þ
This condition is similar to the condition of total reflection occurring in regular media where the electromagnetic waves should be transmitted from a denser medium to a rarer medium. The second restriction comes from the inequality (14), in order to discuss the influence of the different physics parameters on the total reflection phenomenon, we divide the indefinite metamaterial into four classes according to Ref. [12] Cutoff ey lx > 0 lx =lz > 0; Anti-cutoff ey lx < 0 lx =lz < 0; Never cutoff ey lx > 0 lx =lz < 0; Always cutoff ey lx < 0 lx =lz > 0:
ð17Þ
Fig. 3. Variation of reflectance with incident angle for an indefinite medium (anti-cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 2; lx ¼ 1; lz ¼ 1 (solid line), ey ¼ 1; lx ¼ 1; lz ¼ 2 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line), and ey ¼ 3; lx ¼ 1; lz ¼ 2 (dashed dotted line) .
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
Fig. 4. Variation of reflectance with incident angle for an indefinite medium (never-cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2 (solid line), ey ¼ 2; lx ¼ 1; lz ¼ 1 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line).
Fig. 5. Variation of reflectance with incident angle for an indefinite medium (always-cutoff medium). e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2. For any incident angle, the reflectance is always unit.
251
the above condition (Eq. (18)). When the incident angle h is smaller than the critical angle hc, the total reflection can occur. This phenomenon is called anomalous total reflection. Moreover, if ey lz > e1 l1 , the critical angle hc will be equal to p/2. In this case, omnidirectional reflection will occur, and for any incident angles, the TE polarized incident waves will be total reflected. This anomalous total reflection cannot occur when the second medium is a conventional medium. It can be found from Fig. 3 that in the anti-cutoff medium, the Brewster angle can be larger than the critical angle for some values of the constitutive parameters, which is different from what we have well-accepted in classical electrodynamics. For the never cutoff medium (ey lx > 0; lx =lz < 0Þ, according to the inequality (14), the total reflection phenomenon cannot exist for any incident angle because k2z is always real, and some electromagnetic waves can always propagate into the second medium. We give the example in Fig. 4. For the always cutoff medium (ey lx < 0; lx =lz > 0Þ, according to the inequality (14), k2z is always imaginary. So the total reflection phenomenon can occur for any incident angle. This is the so-called omnidirectional reflection, and an example has been given in Fig. 5. Table 1 summarizes the existence conditions for the total reflection and the critical angle for TE waves at the interface separating an isotropic regular medium and an indefinite metamaterial with the form of Eq. (1). For the TM waves, the critical angle is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ex l y hc ¼ sin1 ð20Þ ðex =ez Þe1 l1 pffiffiffiffiffiffiffiffi and the cutoff vector is k c ¼ k 0 ez ly . The existence conditions for the total reflection and critical angle have been summarized in Table 2.
Table 1 The conditions for occurrence of total reflection of TE modes Media conditions
Evanescent
Critical angle
Total reflection condition
Cutoff Anti-cutoff
ey lx > 0; lx =lz > 0 ey lx < 0; lx =lz < 0
kx > kc kx < kc
Never cutoff Always cutoff
ey lx > 0; lx =lz < 0 ey lx < 0; lx =lz > 0
Non-existence All real kx
hc hc p/2 Non-existence Non-existence
h > hc ; 0 < ey lz < e1 l1 , normal h < hc ; 0 < ey lz < e1 l1 , anomalous ey lz > e1 l1 , omnidirectional Non-existence Always, omnidirectional
Table 2 The conditions for occurrence of total reflection of TM modes Media conditions
Evanescent
Critical angle
Total reflection condition
Cutoff Anti-cutoff
ex ly > 0; ex =ez > 0 ex ly < 0; ex =ez < 0
kx > kc kx < kc
Never cutoff Always cutoff
ex ly > 0; ex =ez < 0 ex ly < 0; ex =ez > 0
Non-existence All real kx
hc hc p/2 Non-existence Non-existence
h > hc ; 0 < ez ly < e1 l1 , normal h < hc ; 0 < ez ly < e1 l1 , anomalous ez ly > e1 l1 , omnidirectional Non-existence Always, omnidirectional
252
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
When the optical axis of the indefinite metamaterial is parallel to the interface, the phenomenon of total reflection and the existence conditions can be analyzed similarly. The analytical treatment is omitted here for the sake of brevity. 4. Frustrated total reflection of the slab configuration Next we analyze another case, where the indefinite metamaterial is confined in a space region of thickness d in the propagation direction. Total internal reflection occurs when light comes from the first medium under some conditions which have been discussed in Section 3. Although the energy of the wave is totally reflected, there exists an evanescent wave with the electromagnetic fields decaying exponentially away from the interface in the second medium [21]. If a third medium with a large enough refractive-index is placed behind the sufficiently thin second layer, one can recover propagating waves in the third medium and has non-vanishing transmittance. But under certain condition, the total reflection can still occur, and we call this total reflection as frustrated total reflection. Here we qualitatively analyze this situation and obtain the condition for frustrated total reflection. The geometrical structure considered in this section is given in Fig. 6. We consider an indefinite metamaterial slab which separates two semi-infinite media. The two semi-infinite media are isotropic PIM, one of which is characterized by permittivity e1 and permeability l1 (Medium 1) and the other is e2 and l2 (Medium 3). The indefinite metamaterial is denoted as e and l (Medium 2), and the thickness of the slab is d. In this section, we only consider the TE waves when the optical axis of the indefinite metamaterial is normal to the interface. The other cases can be disposed similarly. For the slab configuration, the reflection coefficient is found to be rslab ¼
r12 þ r23 e2u ; 1 þ r12 r23 e2u
ð21Þ
where u ¼ ik 2z d; r12 is the reflection coefficient when the light goes from semi-infinite medium 1 (Medium 1) to the finite indefinite metamaterial (Medium 2), and r23 is the reflection coefficient when the light goes from finite indefi-
Fig. 7. Dependence of reflectance jrslab j2 on the thickness of indefinite medium for different incident angle. The physical parameters are e1 ¼ l1 ¼ 2 , e2 ¼ l2 ¼ 2 and ey ¼ 1, lx ¼ 1; lz ¼ 2.
nite metamaterial (Medium 2) to the semi-infinite medium 3 (Medium 3). And they can be written as lz k 1z l1 k 2z ; lz k 1z þ l1 k 2z l k 2z lz k 3z r23 ¼ 2 ; l2 k 2z þ lz k 3z
r12 ¼
ð22Þ
where k 1z ; k 2z and k3z are the normal components of the wave vector in medium 1, medium 2 and medium 3, respectively. If medium 1 and medium 3 are the same as medium 1, then the reflection coefficients can be simplified as rslab ¼
r12 ðe2u 1Þ : r212 e2u 1
ð23Þ
The relation between reflectance and thickness d has been given in Fig. 7. For simplicity, we only consider the situation where medium 1 and medium 3 are occupied by the same medium, if medium 1 and medium 3 are different, the results are similar. From Fig. 7, we find that if the interface of medium 1 and medium 2 satisfies the condition of total reflection ðhc ¼ 45 Þ, but the thickness of medium 2 is finite, then the reflectance depends on the thickness of indefinite metamaterial. When the thickness d k, the total reflection always occurs, but when d 6 k, for some inci2 dent angle, jrslab j < 1 at the interface of medium 1 and medium 2, the incident energy cannot any more totally flow back into medium 1 and some of the energy will tunnel the indefinite metamaterial and propagate into medium 3. This phenomenon is called photon tunneling. Furthermore, the thickness of the total reflection occurring in indefinite metamaterial becomes thinner as the incident angle increases. 5. Conclusion
Fig. 6. The slab configuration of total internal reflection. The indefinite medium has a finite thickness of d, and its optical axis is the z-axis.
In this paper, we present a detailed investigation on the problem of total reflection at the interface separating an isotropic conventional medium and an indefinite metamaterial when electromagnetic wave is incident on the inter-
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
face from the side of isotropic medium. We have obtained the existence conditions for total reflection and the critical angle. We find that different total reflection phenomena can be observed for different physics parameters. Moreover, we have predicted the existence of anomalous total reflection and omnidirectional total reflection. If the indefinite metamaterial is finite and the thickness is thin enough, the photon tunneling phenomenon will occur. Such exotic phenomena should have beneficial applications in spectroscopy, imaging devices and waveguide. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 10674045 and 10576012), the Program for New Century Excellent Talents in University and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20040532005). References [1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966.
253
[4] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [5] D.R. Smith, N. Kroll, Phys. Rev. Lett. 85 (2000) 2933. [6] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyl’en, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. [7] V.M. Shalaev, W. Cai, U.K. Chettiar, H. Yuan, A.K. Sarychev, V.P. Drachev, A.V. Kildishev, Opt. Lett. 30 (2005) 3356. [8] A. Alu, N. Engheta, IEEE Trans. Antennas Propagat. 51 (2003) 2558. [9] M.A. Antoniades, G.V. Eleftheriades, IEEE Antennas Wireless Propagat Lett. 2 (2003) 103. [10] I.V. Shadrivov, A.A. Sukhorukov, Y.S. Kivshar, Phys. Rev. E 67 (2003) 0576021. [11] A.C. Peacock, N.G.R. Broderick, Opt. Express. 11 (2003) 2502. [12] D.R. Smith, D. Schurig, Phys. Rev. Lett. 90 (2003) 077405. [13] D. Schurig, D.R. Smith, Appl. Phys. Lett. 82 (2003) 2215. [14] L.B. Hu, S.T. Chui, Phys. Rev. B 66 (2002) 085108. [15] N.H. Shen, Q. Wang, J. Chen, Y.-X. Fan, J. Ding, H.-T. Wang, Y. Tian, J. Opt. Soc. Am. B 23 (2006) 904. [16] M. Born, E. Wolf, Principles of Optics, seventh(expanded) ed., Cambridge University Press, Cambridge, 1998. [17] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectroscopy, Wiley, New York, 1986. [18] R.C. Reddick, R.J. Warmack, D.W. Chilcott, S.L. Sharp, T.L. Ferrell, Rev. Sci. Instrum. 61 (1990) 3669. [19] J.L. Pan, H.K.H. Choy, C.G. Fonstad Jr., IEEE Trans. Electron Dev. 47 (2000) 241. [20] H.X. Da, C. Xu, Z.Y. Li, G. Kraftmakher, Phys. Rev. E 71 (2005) 066612. [21] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988.
Total reflection of electromagnetic waves propagating from an isotropic medium to an indefinite metamaterial Yuanjiang Xiang, Xiaoyu Dai, Shuangchun Wen
*
School of Computer and Communication, Hunan University, Changsha 410082, China Received 16 July 2006; received in revised form 20 November 2006; accepted 3 February 2007
Abstract The existence conditions for total reflection and the corresponding critical angle at the interface separating an isotropic medium and an indefinite metamaterial for TE- and TM-polarized electromagnetic waves are obtained. For different kinds of indefinite metamaterial, there appear different total reflection phenomena. Particularly, the anomalous total reflection in which the incident angle is smaller than the critical angle and the Brewster’s angle can be smaller than the critical angle can occur for anti-cutoff medium. Furthermore, the omnidirectional total reflection exists for the always cutoff medium and anti-cutoff medium. The total reflection depends on the thickness of indefinite metamaterial when the indefinite metamaterial is finite, and the photon tunneling phenomenon can occur when the thickness of indefinite metamaterial is smaller than wavelength. Ó 2007 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Bs; 42.25.Gy Keywords: Indefinite medium; Negative index material; Total reflection; Photon tunneling
1. Introduction The negative index material (NIM) with negative permittivity and negative permeability simultaneously, first introduced by Veselago [1], has recently received much attention in the literatures owing to its very unusual properties [2–5], such as negative refraction index [5] and wave vector antiparallel to the Poynting vector [1]. NIMs were realized primarily in the microwave range. Nowadays, NIMs in the near IR and optical range have also been experimentally demonstrated [6,7]. Several applications have been envisioned for these materials: perfect lens [3], compact-cavity resonators [8], phase shifters [9], and efficient waveguide [10,11]. Recently, the characteristics of electromagnetic waves propagation in anisotropic NIM, even an indefinite metamaterial [12] in which not all the principal elements of the permeability and permittivity ten*
Corresponding author. Tel./fax: +86 731 8823474. E-mail address: [email protected] (S. Wen).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.013
sors have the same sign also have been discussed extensively. Schurig and Smith have outlined a potentially useful application of the indefinite metamaterial as a spatial filtering [13]. Hu and Chui have mainly discussed under what conditions anomalous reflection or refraction can occur at the interface when propagating waves pass from one isotropic regular medium into another uniaxially anisotropic NIM medium [14]. Furthermore, Shen and Wang predicted the existence of a total transmission phenomenon at the interface separating a regular material and an indefinite metamaterial under suitable conditions [15]. In classical electromagnetic field theory, when light travels from the optically dense material to the optically rarer material, the total reflection will occur if the incident angle is larger than the critical angle [16]. This phenomenon has been found applications in attenuated total reflection spectroscopy [17], scanning photon tunneling microscopy [18], and microscale thermophotovoltaic devices [19]. In this paper, we investigate the problem of total reflection of
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
electromagnetic waves at the interface formed by an isotropic regular material and an indefinite metamaterial. We present a detailed analysis on the existence conditions for total reflection for all the possible physical parameters. This total reflection phenomenon is significantly different from that for isotropic materials. This paper is organized as following. We first investigate the propagation characteristic at the interface between an isotropic medium and an indefinite metamaterial in Section 2. Then we discuss the existence conditions for the total reflection and the critical angle when the light goes from the isotropic medium to the indefinite metamaterial in Section 3. In Section 4, we analyze the condition for total reflection when the indefinite metamaterial is finite, and present a numerical result. Finally, in Section 5, we give our conclusion. 2. Propagation characteristics of electromagnetic wave at the interface with indefinite metamaterial The geometry of the considered problem is shown in Fig. 1, where the plane wave of angular frequency x is incident from the medium 1 at an incident angle h into the medium 2 (indefinite metamaterial). The medium 1 is characterized by the permittivity e1 and the permeability l1, and the indefinite metamaterial is characterized by the permittivity e, the permeability l. The interface of the two media is parallel to the xy plane, and the normal direction is the z-axis. To simplify the proceeding analysis, we assume the anisotropic permittivity and permeability tensors can be simultaneously diagonalizable, then e and l tensors can be denoted as the following forms: 0 1 0 1 ex 0 0 lx 0 0 B C B C e ¼ @ 0 ey 0 A; l ¼ @ 0 ly 0 A: ð1Þ 0 0 ez 0 0 lz In the subsequent analysis, we assume that the indefinite metamaterial is optically uniaxial, and the optical axis (zaxis) is normal to the interface, then ex ¼ ey 6¼ ez and lx ¼ ly 6¼ lz . Without loss of generality, we assume that the wave vector locates at the xz plane, and the incident electric field is E ¼ ^y E0 eiðkx xþk1z zxtÞ , where kx and k1z are x and z components of the incident wave vector. We first discuss the TE
249
modes, and the TM modes can be treated similarly. For the TE waves, the dispersion relation in the isotropic regular medium is k 2x þ k 21z ¼ e1 l1 ðx2 =c2 Þ;
ð2Þ
and for the indefinite metamaterial, it is k 2x k2 x2 þ 2z ¼ 2 : ey lz ey lx c
ð3Þ
Thus the normal wave vector component in the indefinite metamaterial satisfies, k 22z ¼ ðx2 =c2 Þ½ey lx ðlx =lz Þe1 l1 sin2 h:
ð4Þ
In the absence of loss, the sign of k 22z can be used to distinguish the nature of the plane-wave solutions. k 22z > 0 corresponds to real valued k2z and propagating solutions. k 22z < 0 corresponds to imaginary valued k2z and exponentially growing or decaying (evanescent) solutions. It is noted that the sign of the wave vector k2z must ensure the Poynting vector inside the indefinite metamaterial to point away from the interface between the incident and indefinite metamaterial, and according to reference [20], k2z and lx must keep the same sign. The reflection coefficients and transmission coefficients can be found by considering the boundary conditions for the electric field and magnetic field at the interface lx k 1z l1 k 2z ; lx k 1z þ l1 k 2z 2lx k 1z ¼ : lx k 1z þ l1 k 2z
rTE ¼
ð5Þ
tTE
ð6Þ
Similarly, we also easily obtain the reflection coefficients and transmission coefficients for TM waves: ex k 1z e1 k 02z ; ex k 1z þ e1 k 02z 2ex k 1z ¼ ; ex k 1z þ e1 k 02z
rTM ¼
ð7Þ
tTM
ð8Þ
where 2 2 2 k 02 2z ¼ ðx =c Þ½ex ly ðex =ez Þe1 l1 sin h:
ð9Þ
If the optical axis of the indefinite metamaterial is parallel to the interface between the two media, we can also choose the z-axis to be normal to the interface and the x-axis to be along the optical axis, then ex 6¼ ey ¼ ez and lx 6¼ ly ¼ lz . Finally the reflection coefficient and transmission coefficient can be obtained similarly. It is easily found from Eqs. (5)–(9) that the total reflection occurs when 2
RTE ¼ jrTE j ¼ 1
ð10Þ
for TE waves and when RTM ¼ jrTM j2 ¼ 1 Fig. 1. Geometrical structure of the problems. The optical axis of the indefinite metamaterial is z-axis.
ð11Þ
for TM waves. In the conventional medium, at the interface formed by two regular media, when the incident angle
250
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
larger than the critical angle, the z-component of the wave vectors of the refracted waves becomes imaginary and the total reflection occurs. In contrast, if the interface is associated with an indefinite metamaterial, the situation should be quite different because the permeability or the permittivity of the two media across the interface also could be different. In next section, we detailedly analyze the conditions of the total reflection occurrence and the conditions of critical angle. 3. The condition for occurrence of total reflection According to Eqs. (10) and (11), it can be found that there may have a value of the wave vector k2z, which can make the reflection coefficient RTE or RTM equal to be 1. After some algebra derivations, k2z must satisfy the following condition: k 22z ¼ 0: Hence, the critical angle can be obtained, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ey lx hc ¼ sin1 : ðlx =lz Þe1 l1
ð12Þ
For the cutoff medium, the total reflection can occur under some conditions, according to the inequality (14), the condition is h > hc :
ð18Þ
This condition is the same as the condition for total reflection phenomenon in the conventional medium, i.e., that the incident angle must be larger than the critical angle, and it is the normal total reflection. We have given an example in Fig. 2. Furthermore, we can find that the Brewster’s angle is existent under some conditions even for the TE waves, and that the Brewster’s angle is smaller than the critical angle. For the anti-cutoff medium, as the physical parameters ey lx ; lx =lz are simultaneously negative, thus from the inequality (14) we can find that if the following condition is satisfied: h < hc
ð19Þ
then k2z will be imaginary and the incident waves can be totally reflected. This condition is obviously different from
ð13Þ
The corresponding value of kx for which k 22z ¼ 0 is the cutoff wave vector separating propagating from evanescent solutions, we denote it as kc. From Eq. (4), this value is pffiffiffiffiffiffiffiffi k c ¼ k 0 ey l z . Then if k 22z < 0, the total reflections are always satisfied, viz., k2z becomes an imaginary number, so ey lx ðlx =lz Þe1 l1 sin2 h < 0:
ð14Þ
Next we discuss the existence possibility of total reflection for different physical parameters. The first restriction of critical angle comes from the span of sin2 hc , thus the following inequality set must be satisfied: 2
0 < sin hc < 1:
ð15Þ
The above inequality is, in fact, equivalent to the following inequality: 0 < ey lz < e1 l1 :
Fig. 2. Variation of reflectance with incident angle for an indefinite medium (cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2 (solid line), ey ¼ 2; lx ¼ 1; lz ¼ 1 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line).
ð16Þ
This condition is similar to the condition of total reflection occurring in regular media where the electromagnetic waves should be transmitted from a denser medium to a rarer medium. The second restriction comes from the inequality (14), in order to discuss the influence of the different physics parameters on the total reflection phenomenon, we divide the indefinite metamaterial into four classes according to Ref. [12] Cutoff ey lx > 0 lx =lz > 0; Anti-cutoff ey lx < 0 lx =lz < 0; Never cutoff ey lx > 0 lx =lz < 0; Always cutoff ey lx < 0 lx =lz > 0:
ð17Þ
Fig. 3. Variation of reflectance with incident angle for an indefinite medium (anti-cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 2; lx ¼ 1; lz ¼ 1 (solid line), ey ¼ 1; lx ¼ 1; lz ¼ 2 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line), and ey ¼ 3; lx ¼ 1; lz ¼ 2 (dashed dotted line) .
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
Fig. 4. Variation of reflectance with incident angle for an indefinite medium (never-cutoff medium) for e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2 (solid line), ey ¼ 2; lx ¼ 1; lz ¼ 1 (dashed line), ey ¼ 3; lx ¼ 1; lz ¼ 1 (dotted line).
Fig. 5. Variation of reflectance with incident angle for an indefinite medium (always-cutoff medium). e1 ¼ l1 ¼ 2, and ey ¼ 1; lx ¼ 1; lz ¼ 2. For any incident angle, the reflectance is always unit.
251
the above condition (Eq. (18)). When the incident angle h is smaller than the critical angle hc, the total reflection can occur. This phenomenon is called anomalous total reflection. Moreover, if ey lz > e1 l1 , the critical angle hc will be equal to p/2. In this case, omnidirectional reflection will occur, and for any incident angles, the TE polarized incident waves will be total reflected. This anomalous total reflection cannot occur when the second medium is a conventional medium. It can be found from Fig. 3 that in the anti-cutoff medium, the Brewster angle can be larger than the critical angle for some values of the constitutive parameters, which is different from what we have well-accepted in classical electrodynamics. For the never cutoff medium (ey lx > 0; lx =lz < 0Þ, according to the inequality (14), the total reflection phenomenon cannot exist for any incident angle because k2z is always real, and some electromagnetic waves can always propagate into the second medium. We give the example in Fig. 4. For the always cutoff medium (ey lx < 0; lx =lz > 0Þ, according to the inequality (14), k2z is always imaginary. So the total reflection phenomenon can occur for any incident angle. This is the so-called omnidirectional reflection, and an example has been given in Fig. 5. Table 1 summarizes the existence conditions for the total reflection and the critical angle for TE waves at the interface separating an isotropic regular medium and an indefinite metamaterial with the form of Eq. (1). For the TM waves, the critical angle is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ex l y hc ¼ sin1 ð20Þ ðex =ez Þe1 l1 pffiffiffiffiffiffiffiffi and the cutoff vector is k c ¼ k 0 ez ly . The existence conditions for the total reflection and critical angle have been summarized in Table 2.
Table 1 The conditions for occurrence of total reflection of TE modes Media conditions
Evanescent
Critical angle
Total reflection condition
Cutoff Anti-cutoff
ey lx > 0; lx =lz > 0 ey lx < 0; lx =lz < 0
kx > kc kx < kc
Never cutoff Always cutoff
ey lx > 0; lx =lz < 0 ey lx < 0; lx =lz > 0
Non-existence All real kx
hc hc p/2 Non-existence Non-existence
h > hc ; 0 < ey lz < e1 l1 , normal h < hc ; 0 < ey lz < e1 l1 , anomalous ey lz > e1 l1 , omnidirectional Non-existence Always, omnidirectional
Table 2 The conditions for occurrence of total reflection of TM modes Media conditions
Evanescent
Critical angle
Total reflection condition
Cutoff Anti-cutoff
ex ly > 0; ex =ez > 0 ex ly < 0; ex =ez < 0
kx > kc kx < kc
Never cutoff Always cutoff
ex ly > 0; ex =ez < 0 ex ly < 0; ex =ez > 0
Non-existence All real kx
hc hc p/2 Non-existence Non-existence
h > hc ; 0 < ez ly < e1 l1 , normal h < hc ; 0 < ez ly < e1 l1 , anomalous ez ly > e1 l1 , omnidirectional Non-existence Always, omnidirectional
252
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
When the optical axis of the indefinite metamaterial is parallel to the interface, the phenomenon of total reflection and the existence conditions can be analyzed similarly. The analytical treatment is omitted here for the sake of brevity. 4. Frustrated total reflection of the slab configuration Next we analyze another case, where the indefinite metamaterial is confined in a space region of thickness d in the propagation direction. Total internal reflection occurs when light comes from the first medium under some conditions which have been discussed in Section 3. Although the energy of the wave is totally reflected, there exists an evanescent wave with the electromagnetic fields decaying exponentially away from the interface in the second medium [21]. If a third medium with a large enough refractive-index is placed behind the sufficiently thin second layer, one can recover propagating waves in the third medium and has non-vanishing transmittance. But under certain condition, the total reflection can still occur, and we call this total reflection as frustrated total reflection. Here we qualitatively analyze this situation and obtain the condition for frustrated total reflection. The geometrical structure considered in this section is given in Fig. 6. We consider an indefinite metamaterial slab which separates two semi-infinite media. The two semi-infinite media are isotropic PIM, one of which is characterized by permittivity e1 and permeability l1 (Medium 1) and the other is e2 and l2 (Medium 3). The indefinite metamaterial is denoted as e and l (Medium 2), and the thickness of the slab is d. In this section, we only consider the TE waves when the optical axis of the indefinite metamaterial is normal to the interface. The other cases can be disposed similarly. For the slab configuration, the reflection coefficient is found to be rslab ¼
r12 þ r23 e2u ; 1 þ r12 r23 e2u
ð21Þ
where u ¼ ik 2z d; r12 is the reflection coefficient when the light goes from semi-infinite medium 1 (Medium 1) to the finite indefinite metamaterial (Medium 2), and r23 is the reflection coefficient when the light goes from finite indefi-
Fig. 7. Dependence of reflectance jrslab j2 on the thickness of indefinite medium for different incident angle. The physical parameters are e1 ¼ l1 ¼ 2 , e2 ¼ l2 ¼ 2 and ey ¼ 1, lx ¼ 1; lz ¼ 2.
nite metamaterial (Medium 2) to the semi-infinite medium 3 (Medium 3). And they can be written as lz k 1z l1 k 2z ; lz k 1z þ l1 k 2z l k 2z lz k 3z r23 ¼ 2 ; l2 k 2z þ lz k 3z
r12 ¼
ð22Þ
where k 1z ; k 2z and k3z are the normal components of the wave vector in medium 1, medium 2 and medium 3, respectively. If medium 1 and medium 3 are the same as medium 1, then the reflection coefficients can be simplified as rslab ¼
r12 ðe2u 1Þ : r212 e2u 1
ð23Þ
The relation between reflectance and thickness d has been given in Fig. 7. For simplicity, we only consider the situation where medium 1 and medium 3 are occupied by the same medium, if medium 1 and medium 3 are different, the results are similar. From Fig. 7, we find that if the interface of medium 1 and medium 2 satisfies the condition of total reflection ðhc ¼ 45 Þ, but the thickness of medium 2 is finite, then the reflectance depends on the thickness of indefinite metamaterial. When the thickness d k, the total reflection always occurs, but when d 6 k, for some inci2 dent angle, jrslab j < 1 at the interface of medium 1 and medium 2, the incident energy cannot any more totally flow back into medium 1 and some of the energy will tunnel the indefinite metamaterial and propagate into medium 3. This phenomenon is called photon tunneling. Furthermore, the thickness of the total reflection occurring in indefinite metamaterial becomes thinner as the incident angle increases. 5. Conclusion
Fig. 6. The slab configuration of total internal reflection. The indefinite medium has a finite thickness of d, and its optical axis is the z-axis.
In this paper, we present a detailed investigation on the problem of total reflection at the interface separating an isotropic conventional medium and an indefinite metamaterial when electromagnetic wave is incident on the inter-
Y. Xiang et al. / Optics Communications 274 (2007) 248–253
face from the side of isotropic medium. We have obtained the existence conditions for total reflection and the critical angle. We find that different total reflection phenomena can be observed for different physics parameters. Moreover, we have predicted the existence of anomalous total reflection and omnidirectional total reflection. If the indefinite metamaterial is finite and the thickness is thin enough, the photon tunneling phenomenon will occur. Such exotic phenomena should have beneficial applications in spectroscopy, imaging devices and waveguide. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 10674045 and 10576012), the Program for New Century Excellent Talents in University and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20040532005). References [1] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [2] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966.
253
[4] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [5] D.R. Smith, N. Kroll, Phys. Rev. Lett. 85 (2000) 2933. [6] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyl’en, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. [7] V.M. Shalaev, W. Cai, U.K. Chettiar, H. Yuan, A.K. Sarychev, V.P. Drachev, A.V. Kildishev, Opt. Lett. 30 (2005) 3356. [8] A. Alu, N. Engheta, IEEE Trans. Antennas Propagat. 51 (2003) 2558. [9] M.A. Antoniades, G.V. Eleftheriades, IEEE Antennas Wireless Propagat Lett. 2 (2003) 103. [10] I.V. Shadrivov, A.A. Sukhorukov, Y.S. Kivshar, Phys. Rev. E 67 (2003) 0576021. [11] A.C. Peacock, N.G.R. Broderick, Opt. Express. 11 (2003) 2502. [12] D.R. Smith, D. Schurig, Phys. Rev. Lett. 90 (2003) 077405. [13] D. Schurig, D.R. Smith, Appl. Phys. Lett. 82 (2003) 2215. [14] L.B. Hu, S.T. Chui, Phys. Rev. B 66 (2002) 085108. [15] N.H. Shen, Q. Wang, J. Chen, Y.-X. Fan, J. Ding, H.-T. Wang, Y. Tian, J. Opt. Soc. Am. B 23 (2006) 904. [16] M. Born, E. Wolf, Principles of Optics, seventh(expanded) ed., Cambridge University Press, Cambridge, 1998. [17] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectroscopy, Wiley, New York, 1986. [18] R.C. Reddick, R.J. Warmack, D.W. Chilcott, S.L. Sharp, T.L. Ferrell, Rev. Sci. Instrum. 61 (1990) 3669. [19] J.L. Pan, H.K.H. Choy, C.G. Fonstad Jr., IEEE Trans. Electron Dev. 47 (2000) 241. [20] H.X. Da, C. Xu, Z.Y. Li, G. Kraftmakher, Phys. Rev. E 71 (2005) 066612. [21] P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988.