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Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer
Physics Letters A 351 (2006) 391–397 www.elsevier.com/locate/pla
Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer Shuming Wang a , Chaojun Tang a , Tao Pan c , Lei Gao b,a,∗ a Department of Physics, Suzhou University, Suzhou 215006, China 1 b CCAST (World Laboratory), PO Box 8730, Beijing 100080, China c Department of Physics, University of Science and Technology of Suzhou, Suzhou 215009, China
Received 25 September 2005; accepted 9 November 2005 Available online 18 November 2005 Communicated by A.R. Bishop
Abstract Using effective medium approximation (EMA) method, we investigate the effective permittivity and permeability of the bilayer composed of negative-permittivity layer and negative-permeability layer. Analytic expressions of all six elements of the effective permittivity and permeability tensors are derived. It is found that these elements have different signs and values at different frequency regions. To one’s interest, negative refraction can be realized in the bilayer, which may provide an easier way to fabricate negatively refractive materials. Moreover, we discuss the EMA method with both small |klz |dl and large |klz |dl . In addition, we also investigate the excitation of the surface polaritons at the interfaces of the bilayer. Like negatively refractive materials, the bilayer can also amplify evanescent waves owing to the excitation of the surface polaritons at the interfaces, which can confirm the equivalence between the bilayer and a slab of negatively refractive material. 2005 Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 41.20.Jb; 73.20.Mf; 78.66.-w
1. Introduction In 1968, Veselago [1] proposed the concept of left-handed materials (having different names in many papers, for example, negatively refractive material) with both negative permittivity and negative permeability, and predicted some peculiar electromagnetic properties such as the reversal of Doppler shift for radiation, the reversal of Cherenkov radiation and the negative refraction. But, this materials did not arose extensive interests until recent years, because this kind of material is not available naturally. In 1999, Pendry et al. [2] successfully realized the negative permeability in split-ring resonator. Subsequently, 3D electromagnetic artificial materials (an array of resonant cell consisting of thin wire strip and split-ring resonator) were realized to synthesize negatively refractive materials, thus, the negative refraction predicted a long time ago by Veselago was first realized [3–5]. In 2000, Pendry [6] proposed that a flat slab of homogeneous negatively refractive material can focus an object in front of it. Not only far field consisting of propagating plane waves but also the near field consisting of evanescent plane waves can be focused through such a slab. This enables one to circumvent the diffraction limited resolution and allows for sub-wavelength imaging. Thereafter, more and more attentions were paid to negatively refractive materials. Due to the absence of negatively refractive material naturally, many different ways [7–9] were proposed to fabricate it. Fredkin et al. [8] showed that a layered material, with alternating layers having negative permittivity and negative permeability, is in many ways similar with a negatively refractive material. Lakhtakia et al. [9] also analyzed the equivalence between a similar structure * Corresponding author.
E-mail address: [email protected] (L. Gao). 1 Mailing address.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.11.025
392
S. Wang et al. / Physics Letters A 351 (2006) 391–397
and negatively refractive material. Both of them want to use layered materials to make negatively refractive materials because the method may have some advantages over other methods [8]. In this Letter, using transfer matrix method [10], we will investigate the equivalence between the bilayer made of negative-permittivity layer and negative-permeability layer and the negatively refractive material. We will show that with the aid of EMA method, the bilayer can be equivalent to a uniaxial negatively refractive material by calculating the effective permittivity and permeability of the bilayer. To further confirm the equivalence, we will also investigate the amplification of the evanescent wave due to the excitation of the surface polaritons at the interfaces of the bilayer structure similar with the negatively refractive materials. The rest of the Letter is organized as follows. In Section 2, we give out the derivation of the parameters of effective permittivity and permeability and the analysis of the refraction and cutoff properties of the bilayer. In Section 3, we investigate the amplification of the evanescent wave due to the excitation of the surface polaritons in the bilayer. Finally, the summary is given in Section 4. 2. Effective medium approximation (EMA) method 2.1. Equivalence of bilayer The geometry considered here is shown in Fig. 1. We assume the surrounding environment to be vacuum, namely εs = 1, µs = 1. Without loss of generality, we take a two-dimensional light beam of transverse electric (TE) polarization (transverse magnetic (TM) polarization can be discussed in a similar way) with its wave vector in the x–z plane. The electric and magnetic field in the system can be expressed as El = Al eiklz (z−zl−1 ) + Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ey , klz iklz (z−zl−1 ) kx iklz (z−zl−1 ) Hl = − (1) Al e Al e − Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ex + + Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ez , ωµl ωµl where zl = zl−1 + dl is the position of the lth interface with z1 = 0, d1 = 0 and d4 = 0, and l refers to the layer number, l = 1, 2, 3, 4. Note that B4 should be zero, since no backward waves exist in the last layer. The z-component of wave vector k in the bilayer structure has the form as sin2 (θ ) ω √ √ klz = εl µl 1 − (2) , εl µl c where θ refers to the incident angle of the electromagnetic wave, with sin θ = kx /k0 (k0 ≡ ω/c). For the evanescent waves, kx /k0 can be larger than one. Using the transfer matrix method, we have the transfer matrices of the two layers of the bilayer structure as follows, cos(klz dl ) − kµlzl sin(klz dl ) Tl = (3) . klz cos(klz dl ) µl sin(klz dl ) In the present Letter, the electric permittivities and the magnetic permeabilities of the two layer are assumed to be isotropic and have the following forms [11], ε2 = 3,
µ2 = 1 −
F ω2 , ω2 − ω02
and
ε3 = 1 −
ωp2 ω2
,
µ3 = 1,
Fig. 1. The geometry of the bilayer structure considered with d2 = d3 = 0.003 mm.
(4)
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where we have the plasma angular frequency ωp = 10 × 109 rad/s, the resonant angular frequency ω0 = 4 × 109 rad/s and the constant F = 0.56, and we can see that layer 2 and layer 3 have respectively negative permeability µ2 and negative permittivity ε3 simultaneously in the angular frequency region of [4 × 109 rad/s, 6 × 109 rad/s]. It is evident that the negative permittivity and negative permeability are realized in the GHz region. Therefore, the product of the wave vector klz and the thicknesses of the layers dl generally satisfies the condition |klz |dl 1, where |klz | is the module value of klz . As a result, we have sin(klz dl ) ≈ klz dl , and cos(klz dl ) ≈ 1. Then we can use the EMA method to get the effective permittivity and permeability of the bilayer structure. The total transfer matrix of the bilayer can be 1 −(µ2 f2 + µ3 f3 )d Ttotal = T2 T3 = (5) , 2 ((ε2 f2 + ε3 f3 ) ωc2 − ( µf22 + µf33 )kx2 )d 1 where d = d1 + d2 , f1 = d1 /d, f2 = d2 /d and for the continuity of the transverse wave vectors, k1x = k2x = kx . On the other hand, since the layered structure has an anisotropic electromagnetic property, we can assume that the effective slab has the effective permittivity and permeability to be anisotropic and have the diagonalizable forms as follows [12]. εeffx µeffx 0 0 0 0 ¯ε¯ eff = ¯ (6) , and µ¯ eff = . 0 0 0 εeffy 0 µeffy 0 0 εeffz 0 0 µeffz With the above approximation, the effective transfer matrix of the bilayer can be written as effx cos(keffz d) − µkeffz sin(keffz d) 1 −µeffx d = . Teff = k (7) 2 kx2 effz )d 1 (εeffy ωc2 − µeffz sin(k d) cos(k d) effz effz µeffx Comparing Eq. (5) and Eq. (7), we then get the three elements of the effective permittivity and permeability tensors of the bilayer, µeffx = µ1 f1 + µ2 f2 ,
εeffy = ε1 f1 + ε2 f2 ,
µeffz =
1 f1 µ1
+
f2 µ2
.
(8)
As for the TM mode case, we just need to make a substitution of ε ↔ µ. Then the other three effective elements can be obtained εeffx = ε1 f1 + ε2 f2 ,
µeffy = µ1 f1 + µ2 f2 ,
εeffz =
1 f1 ε1
+
f2 ε2
.
(9)
t , and µ t Obviously, in the case considered, we have εeffx = εeffy = εeff effx = µeffy = µeff , so that the bilayer composed of one negative permittivity layer and one negative permeability layer makes a uniaxial material. Different from the equivalence in Ref. [9], in the present Letter, we obtain all the elements of the effective permittivity tensor and permeability tensor. We can see that these effective elements are independent on the incident angle, which is totally different from the angle-dependent results derived in the previous papers, and may have more physical meanings. Substituting Eq. (4) into Eqs. (8) and (9), we get the analytic expressions of all the parameters of the effective permittivity and permeability for d1 = d2 ,
50 3ω2 − 300 0.28ω2 z , µteff = 1 − 2 , , εeff = εeffz = 2 2 ω 2ω − 50 ω − 16 In the uniaxial material, the z-component of the wave vector has the form as t εeff =2−
ω 2 µt 2 − z kx , µ c2 for TE mode, and kz2 = ε t µt
µzeff = µeffz =
0.44ω2 − 16 . 0.72ω2 − 16
(10)
(11)
ω2 ε t 2 − z kx , (12) ε c2 for the TM mode. With these dispersion relations, one can obtain the group velocity vg ≡ ∇k ω(k). The relationship between the directions of the group velocity and wave vector determines the refraction property of the bilayer. Moreover, with different values and signs of the effective parameters, one can divide the cutoff conditions into four kinds: normal-cutoff (kz change from real to imaginary when the incident angle increases), anti-cutoff (opposite to normal cutoff), never-cutoff (for any incident angle, kz is always real), and always-cutoff (for any incident angle, kz is always imaginary) [12]. For the effectively uniaxial material, the effective elements are plotted in Fig. 2. The figure is divided into six regions according to the different signs of the elements of the effective permittivity and permeability tensors. In Table 1 we list the refraction properties in the third column, where ‘+’ refers to positive refraction and ‘−’ to the negative refraction. And the cutoff conditions for both TE and TM mode waves in the six regions are listed in the fourth column. It is the first time to our best knowledge that for the long wave approximation the equivalence of the bilayer is analyzed in details. kz2 = ε t µt
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Fig. 2. The corresponding elements of the effective permittivity and permeability tensors as a function of angular frequency. Table 1 Effective parameters I II III IV V VI
µt µt µt µt µt µt
> 0, µz < 0, µz > 0, µz > 0, µz > 0, µz > 0, µz
> 0, εt > 0, εt < 0, εt < 0, εt > 0, εt > 0, εt
< 0, εz < 0, εz < 0, εz > 0, εz > 0, εz > 0, εz
Cutoff condition >0 >0 >0 <0 <0 >0
–/+ – + + + +
always-cutoff for TE mode, anti-cutoff for TM mode never-cutoff for both TE and TM mode anti-cutoff for both TE and TM mode never-cutoff for both TE and TM mode normal-cutoff for TE mode, never-cutoff for TM mode normal-cutoff for both TE and TM mode
In region I, TE mode is negative refraction (since the electromagnetic wave is always cutoff in this region, the application of the negative refraction in this region is limited), and TM mode is positive refraction. In the table, we can see that in the region II, the bilayer can realize negative refraction, e.g., in the angular frequency region of [4 × 109 rad/s, 4.714 × 109 rad/s]. This indicates a potential application in fabricating of negatively refractive materials. 2.2. Discussion on the EMA method This EMA method not only can be used to investigate the bilayer structure, but also can be used to study the periodic structure, as long as |klz |dl 1. In fact, the EMA method can be used in the condition even when |klz |dl ≈ 0.5 [13]. The results of the EMA method can very well describe the periodic structures consists of single negative materials. According to the Bloch Theorem, the periodic structures have the dispersion relation as follows, 1 µ3 k2z µ2 k3z + cos(qd) = cos(k2z d2 ) cos(k3z d3 ) − (13) sin(k2z d2 ) sin(k3z d3 ). 2 µ2 k3z µ3 k2z 2 d 2 , sin(k If we take both cos(k2z d2 ), sin(k3z d3 ) up to the second order, that is cos(k1,2z d1,2 ) = 1 − 12 k1,2z 1,2z d1,2 ) = k1,2z d1,2 . 1,2 2
t ω − After some simple derivation, we can get q 2 = µteff εeff c2
µteff 2 k . µzeff x
t are simultaneously For normal incidence, when µteff and εeff
t have different signs, q 2 is negative, the periodic structure can realize negative refraction. On the other hand, when µteff and εeff negative and the electromagnetic wave cannot propagate in the periodic structure, which indicates a gap. To one’s interest, when |klz |dl is not very large, e.g., |klz |dl < 0.5, the zero–φ gap in Refs. [14,15] just coincides with the gap mentioned above. Thus, the
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boundary positions of the gap can be easily figured out by the results of this Letter. For instance, in Ref. [14], the boundary of the zero–φ gap is calculated to be 4.53557 × 109 and 7.90569 × 109 rad/s which is almost the same as the numerical result, with kd is about 0.4. From this, we can see the usefulness of the EMA method used in this Letter. However, for the large value of |klz |dl , the approximation that sin(klz dl ) ≈ klz dl , and cos(klz dl) ≈ 1 cannot be used. Thus, the EMA method in the present Letter cannot work. Then, to get the effective parameters of the system (bilayer or finite periodic structure), we can use the numerical methods. Here, we show one of these methods and compare it with the method in this paper to further analyze the EMA method. Consider the normal incident case, the total transfer matrix is obtained from the product of all the matrices of the layers. m11 m12 , Ttotal = (M1 M2 )N = (14) m21 m22 in which, N is the number of periods in the periodic structure, and N = 1 refers to the bilayer. To compare Eq. (14) with the first expression in Eq. (7), we can get the effective parameters of the system, z 2 z m12 keff c c arccos(m11 ) 1 keff z t keff (15) = = t . , µteff = − , εeff d ω ω µeff z and µzeff cannot be figured out in the method. Moreover, for The details of the method is dwelt on in Ref. [15]. However, εeff the oblique incidence, i.e., sin2 (θ ) = 0, the effective permittivity and permeability tensors of the system cannot be calculated. One can only give out the effective permittivity and effective permeability, which must be θ -dependent [16]. Actually, for large |klz |dl , the effective permittivity and permeability εeff (µeff ) are not only dependent on ε1 , ε2 (µ1 , µ2 ), εeff (µeff ) is also a function (t,z) (t,z) of µ1 , µ2 (ε1 , ε2 ). Only when the condition |klz |dl 1, the coupling of ε and µ vanishes, then εeff and µeff have the simple and analytic expressions as Eqs. (8) and (9).
3. Amplification of evanescent wave It is well known that negatively refractive materials can amplify evanescent wave due to the excitation of the surface polaritons at the interface between negatively refractive material and positively refractive material [17]. This is one of the most important properties of the negatively refractive materials. To confirm further the equivalence between negatively refractive material and the bilayer made of negative permittivity layer and negative permeability layer, we shall investigate the surface polaritons supported by the bilayer and calculate the transmission coefficient of the evanescent wave. The transmission coefficient [the absolute value of A4 /A1 in Eq. (1)] of evanescent wave can be easily obtained from the transfer matrix method. Using the method in Refs. [12,18,19] and the transfer matrix, we can obtain the dispersion relation of surface polariton supported by this bilayer [20], f (ω, kx ) = cosh(α2 d2 ) cosh(α3 d3 ) + + where α 1 = α4 =
α 2 µ2 + α22 µ23 α22 µ21 + α12 µ22 sinh(α2 d2 ) cosh(α3 d3 ) + 3 2 sinh(α2 d2 ) sinh(α3 d3 ) 2α2 µ1 α1 µ2 2α3 µ2 α2 µ3
α42 µ23 + α32 µ24 cosh(α2 d2 ) cosh(α3 d3 ) = 0, 2α4 µ3 α3 µ4
(16)
ω2 kx2 − 2 εs µs , c
α2 =
ω2 kx2 − 2 ε2 µ2 , c
α3 =
kx2 −
ω2 ε3 µ3 , c2
(17)
and here kx can be larger than k0 . In Fig. 3(a), we plot f (ω, kx ) as a function of frequency for kx /k0 = 2.5. It can be shown that there exist two points of intersection between f (ω, kx ) and the x-axis. This indicates that there are two angular frequencies that satisfy Eq. (16). In other words, for kx /k0 = 2.5, at these two angular frequencies, the surface polaritons can be excited by the bilayer. The transmission coefficient |A4 /A1 | as a function of angular frequency for kx /k0 = 2.5 is shown in Fig. 3(b). Comparing carefully Fig. 3(a) with Fig. 3(b), we can find that the positions of the two points of intersection in Fig. 3(a) are just the positions at which the transmission coefficient diverges. Hence, the excitation of surface polariton can greatly contribute to the amplification of the evanescent wave. And we can see that in the effectively negative refraction region, the bilayer can amplify the evanescent wave. To see the details clearly, we plot the distribution of the electric field of an evanescent wave in the bilayer in Fig. 3(c) near the intersection angular frequency in the negative refraction region. At the first interface between air and the layer 2 (negative permeability), there is an anti-surface polariton, and at the second interface between layer 2 (negative permeability) and layer 3 (negative permittivity), a surface polariton is excited, where the evanescent wave reaches the peak and then falls. Since the layer 3 and air have both positive permeability (in this case, µ3 = 1), for the TE mode wave, the signs of the slope of the electric field at both sides of the third interface are the same, so that no (anti-)surface polariton can be excited. Therefore, we can see that the amplification of the evanescent wave from the bilayer is a total effect of the excitation of the (anti-)surface polaritons at the first and
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(a)
(b)
(c) Fig. 3. The transmission coefficient and f (ω, kx ) as a function of angular frequency are plotted in (a) and (b). The electric field distribution in the bilayer is plotted in (c), for ω = 4.03 × 109 rad/s.
second interfaces, for TE mode wave. For the TM mode wave, the excitation of surface polaritons at the second and third interfaces will contribute to the amplification of the evanescent wave of the bilayer.
4. Summary
In conclusion, using EMA method, we have investigated the equivalence between negatively refractive material and the bilayer made of one negative permittivity layer and one negative permeability layer. We first derive the analytic expressions of all six parameters of the effective permittivity and permeability, under the condition that |klz |dl 1 is satisfied. It is found that the bilayer can be effectively regarded as a uniaxial material and the bilayer can realize negative refraction at the frequency region of [4 × 109 rad/s, 4.714 × 109 rad/s]. And we also analyze the refraction properties and the cutoff condition of the bilayer as well. With the aid of the EMA method, we further investigate the periodic structure when |klz |dl is small, and the case when |klz |dl is large is investigated, too. To confirm the equivalence, we investigate the surface polaritons excited at the interfaces of the bilayer, and find that the bilayer can amplify the evanescent wave indeed as well as the negatively refractive materials.
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Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10204017 and the Natural Science of Jiangsu Province under Grant No. BK2002038. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. R.W. Ziolkowski, IEEE Trans. Antennas Propagation 47 (2003) 1516. J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, C.M. Soukoulis, Phys. Rev. Lett. 91 (2003) 207401. D.R. Fredkin, A. Ron, J. Appl. Phys. 81 (2002) 1753. A. Lakhtakia, C.M. Krowne, Optik 114 (2003) 305. H. Jiang, H. Chen, H. Li, Y. Zhang, S. Zhu, Appl. Phys. Lett. 83 (2003) 5386. R. Ruppin, Phys. Lett. A 277 (2001) 61. D.R. Smith, D. Schuring, Phys. Rev. Lett. 90 (2003) 077405. A.A. Krokhin, E. Reyes, Phys. Rev. Lett. 93 (2004) 023904. L. Wang, H. Chen, S. Zhu, Phys. Rev. B 70 (2004) 245102. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Li, S. Zhu, Phys. Rev. E 69 (2004) 066607. R.X. Wu, J. Appl. Phys. 97 (2005) 076105. X.S. Rao, C.K. Ong, Phys. Rev. B 68 (2003) 113103. L. Zhou, C.T. Chan, Appl. Phys. Lett. 84 (2004) 1444. R. Ruppin, J. Phys.: Condens. Matter 13 (2001) 1811. C.J. Tang, L. Gao, J. Phys.: Condens. Matter 16 (2004) 4743.
Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer Shuming Wang a , Chaojun Tang a , Tao Pan c , Lei Gao b,a,∗ a Department of Physics, Suzhou University, Suzhou 215006, China 1 b CCAST (World Laboratory), PO Box 8730, Beijing 100080, China c Department of Physics, University of Science and Technology of Suzhou, Suzhou 215009, China
Received 25 September 2005; accepted 9 November 2005 Available online 18 November 2005 Communicated by A.R. Bishop
Abstract Using effective medium approximation (EMA) method, we investigate the effective permittivity and permeability of the bilayer composed of negative-permittivity layer and negative-permeability layer. Analytic expressions of all six elements of the effective permittivity and permeability tensors are derived. It is found that these elements have different signs and values at different frequency regions. To one’s interest, negative refraction can be realized in the bilayer, which may provide an easier way to fabricate negatively refractive materials. Moreover, we discuss the EMA method with both small |klz |dl and large |klz |dl . In addition, we also investigate the excitation of the surface polaritons at the interfaces of the bilayer. Like negatively refractive materials, the bilayer can also amplify evanescent waves owing to the excitation of the surface polaritons at the interfaces, which can confirm the equivalence between the bilayer and a slab of negatively refractive material. 2005 Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 41.20.Jb; 73.20.Mf; 78.66.-w
1. Introduction In 1968, Veselago [1] proposed the concept of left-handed materials (having different names in many papers, for example, negatively refractive material) with both negative permittivity and negative permeability, and predicted some peculiar electromagnetic properties such as the reversal of Doppler shift for radiation, the reversal of Cherenkov radiation and the negative refraction. But, this materials did not arose extensive interests until recent years, because this kind of material is not available naturally. In 1999, Pendry et al. [2] successfully realized the negative permeability in split-ring resonator. Subsequently, 3D electromagnetic artificial materials (an array of resonant cell consisting of thin wire strip and split-ring resonator) were realized to synthesize negatively refractive materials, thus, the negative refraction predicted a long time ago by Veselago was first realized [3–5]. In 2000, Pendry [6] proposed that a flat slab of homogeneous negatively refractive material can focus an object in front of it. Not only far field consisting of propagating plane waves but also the near field consisting of evanescent plane waves can be focused through such a slab. This enables one to circumvent the diffraction limited resolution and allows for sub-wavelength imaging. Thereafter, more and more attentions were paid to negatively refractive materials. Due to the absence of negatively refractive material naturally, many different ways [7–9] were proposed to fabricate it. Fredkin et al. [8] showed that a layered material, with alternating layers having negative permittivity and negative permeability, is in many ways similar with a negatively refractive material. Lakhtakia et al. [9] also analyzed the equivalence between a similar structure * Corresponding author.
E-mail address: [email protected] (L. Gao). 1 Mailing address.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.11.025
392
S. Wang et al. / Physics Letters A 351 (2006) 391–397
and negatively refractive material. Both of them want to use layered materials to make negatively refractive materials because the method may have some advantages over other methods [8]. In this Letter, using transfer matrix method [10], we will investigate the equivalence between the bilayer made of negative-permittivity layer and negative-permeability layer and the negatively refractive material. We will show that with the aid of EMA method, the bilayer can be equivalent to a uniaxial negatively refractive material by calculating the effective permittivity and permeability of the bilayer. To further confirm the equivalence, we will also investigate the amplification of the evanescent wave due to the excitation of the surface polaritons at the interfaces of the bilayer structure similar with the negatively refractive materials. The rest of the Letter is organized as follows. In Section 2, we give out the derivation of the parameters of effective permittivity and permeability and the analysis of the refraction and cutoff properties of the bilayer. In Section 3, we investigate the amplification of the evanescent wave due to the excitation of the surface polaritons in the bilayer. Finally, the summary is given in Section 4. 2. Effective medium approximation (EMA) method 2.1. Equivalence of bilayer The geometry considered here is shown in Fig. 1. We assume the surrounding environment to be vacuum, namely εs = 1, µs = 1. Without loss of generality, we take a two-dimensional light beam of transverse electric (TE) polarization (transverse magnetic (TM) polarization can be discussed in a similar way) with its wave vector in the x–z plane. The electric and magnetic field in the system can be expressed as El = Al eiklz (z−zl−1 ) + Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ey , klz iklz (z−zl−1 ) kx iklz (z−zl−1 ) Hl = − (1) Al e Al e − Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ex + + Bl e−iklz (z−zl−1 ) ei(kx x−ωt) ez , ωµl ωµl where zl = zl−1 + dl is the position of the lth interface with z1 = 0, d1 = 0 and d4 = 0, and l refers to the layer number, l = 1, 2, 3, 4. Note that B4 should be zero, since no backward waves exist in the last layer. The z-component of wave vector k in the bilayer structure has the form as sin2 (θ ) ω √ √ klz = εl µl 1 − (2) , εl µl c where θ refers to the incident angle of the electromagnetic wave, with sin θ = kx /k0 (k0 ≡ ω/c). For the evanescent waves, kx /k0 can be larger than one. Using the transfer matrix method, we have the transfer matrices of the two layers of the bilayer structure as follows, cos(klz dl ) − kµlzl sin(klz dl ) Tl = (3) . klz cos(klz dl ) µl sin(klz dl ) In the present Letter, the electric permittivities and the magnetic permeabilities of the two layer are assumed to be isotropic and have the following forms [11], ε2 = 3,
µ2 = 1 −
F ω2 , ω2 − ω02
and
ε3 = 1 −
ωp2 ω2
,
µ3 = 1,
Fig. 1. The geometry of the bilayer structure considered with d2 = d3 = 0.003 mm.
(4)
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where we have the plasma angular frequency ωp = 10 × 109 rad/s, the resonant angular frequency ω0 = 4 × 109 rad/s and the constant F = 0.56, and we can see that layer 2 and layer 3 have respectively negative permeability µ2 and negative permittivity ε3 simultaneously in the angular frequency region of [4 × 109 rad/s, 6 × 109 rad/s]. It is evident that the negative permittivity and negative permeability are realized in the GHz region. Therefore, the product of the wave vector klz and the thicknesses of the layers dl generally satisfies the condition |klz |dl 1, where |klz | is the module value of klz . As a result, we have sin(klz dl ) ≈ klz dl , and cos(klz dl ) ≈ 1. Then we can use the EMA method to get the effective permittivity and permeability of the bilayer structure. The total transfer matrix of the bilayer can be 1 −(µ2 f2 + µ3 f3 )d Ttotal = T2 T3 = (5) , 2 ((ε2 f2 + ε3 f3 ) ωc2 − ( µf22 + µf33 )kx2 )d 1 where d = d1 + d2 , f1 = d1 /d, f2 = d2 /d and for the continuity of the transverse wave vectors, k1x = k2x = kx . On the other hand, since the layered structure has an anisotropic electromagnetic property, we can assume that the effective slab has the effective permittivity and permeability to be anisotropic and have the diagonalizable forms as follows [12]. εeffx µeffx 0 0 0 0 ¯ε¯ eff = ¯ (6) , and µ¯ eff = . 0 0 0 εeffy 0 µeffy 0 0 εeffz 0 0 µeffz With the above approximation, the effective transfer matrix of the bilayer can be written as effx cos(keffz d) − µkeffz sin(keffz d) 1 −µeffx d = . Teff = k (7) 2 kx2 effz )d 1 (εeffy ωc2 − µeffz sin(k d) cos(k d) effz effz µeffx Comparing Eq. (5) and Eq. (7), we then get the three elements of the effective permittivity and permeability tensors of the bilayer, µeffx = µ1 f1 + µ2 f2 ,
εeffy = ε1 f1 + ε2 f2 ,
µeffz =
1 f1 µ1
+
f2 µ2
.
(8)
As for the TM mode case, we just need to make a substitution of ε ↔ µ. Then the other three effective elements can be obtained εeffx = ε1 f1 + ε2 f2 ,
µeffy = µ1 f1 + µ2 f2 ,
εeffz =
1 f1 ε1
+
f2 ε2
.
(9)
t , and µ t Obviously, in the case considered, we have εeffx = εeffy = εeff effx = µeffy = µeff , so that the bilayer composed of one negative permittivity layer and one negative permeability layer makes a uniaxial material. Different from the equivalence in Ref. [9], in the present Letter, we obtain all the elements of the effective permittivity tensor and permeability tensor. We can see that these effective elements are independent on the incident angle, which is totally different from the angle-dependent results derived in the previous papers, and may have more physical meanings. Substituting Eq. (4) into Eqs. (8) and (9), we get the analytic expressions of all the parameters of the effective permittivity and permeability for d1 = d2 ,
50 3ω2 − 300 0.28ω2 z , µteff = 1 − 2 , , εeff = εeffz = 2 2 ω 2ω − 50 ω − 16 In the uniaxial material, the z-component of the wave vector has the form as t εeff =2−
ω 2 µt 2 − z kx , µ c2 for TE mode, and kz2 = ε t µt
µzeff = µeffz =
0.44ω2 − 16 . 0.72ω2 − 16
(10)
(11)
ω2 ε t 2 − z kx , (12) ε c2 for the TM mode. With these dispersion relations, one can obtain the group velocity vg ≡ ∇k ω(k). The relationship between the directions of the group velocity and wave vector determines the refraction property of the bilayer. Moreover, with different values and signs of the effective parameters, one can divide the cutoff conditions into four kinds: normal-cutoff (kz change from real to imaginary when the incident angle increases), anti-cutoff (opposite to normal cutoff), never-cutoff (for any incident angle, kz is always real), and always-cutoff (for any incident angle, kz is always imaginary) [12]. For the effectively uniaxial material, the effective elements are plotted in Fig. 2. The figure is divided into six regions according to the different signs of the elements of the effective permittivity and permeability tensors. In Table 1 we list the refraction properties in the third column, where ‘+’ refers to positive refraction and ‘−’ to the negative refraction. And the cutoff conditions for both TE and TM mode waves in the six regions are listed in the fourth column. It is the first time to our best knowledge that for the long wave approximation the equivalence of the bilayer is analyzed in details. kz2 = ε t µt
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Fig. 2. The corresponding elements of the effective permittivity and permeability tensors as a function of angular frequency. Table 1 Effective parameters I II III IV V VI
µt µt µt µt µt µt
> 0, µz < 0, µz > 0, µz > 0, µz > 0, µz > 0, µz
> 0, εt > 0, εt < 0, εt < 0, εt > 0, εt > 0, εt
< 0, εz < 0, εz < 0, εz > 0, εz > 0, εz > 0, εz
Cutoff condition >0 >0 >0 <0 <0 >0
–/+ – + + + +
always-cutoff for TE mode, anti-cutoff for TM mode never-cutoff for both TE and TM mode anti-cutoff for both TE and TM mode never-cutoff for both TE and TM mode normal-cutoff for TE mode, never-cutoff for TM mode normal-cutoff for both TE and TM mode
In region I, TE mode is negative refraction (since the electromagnetic wave is always cutoff in this region, the application of the negative refraction in this region is limited), and TM mode is positive refraction. In the table, we can see that in the region II, the bilayer can realize negative refraction, e.g., in the angular frequency region of [4 × 109 rad/s, 4.714 × 109 rad/s]. This indicates a potential application in fabricating of negatively refractive materials. 2.2. Discussion on the EMA method This EMA method not only can be used to investigate the bilayer structure, but also can be used to study the periodic structure, as long as |klz |dl 1. In fact, the EMA method can be used in the condition even when |klz |dl ≈ 0.5 [13]. The results of the EMA method can very well describe the periodic structures consists of single negative materials. According to the Bloch Theorem, the periodic structures have the dispersion relation as follows, 1 µ3 k2z µ2 k3z + cos(qd) = cos(k2z d2 ) cos(k3z d3 ) − (13) sin(k2z d2 ) sin(k3z d3 ). 2 µ2 k3z µ3 k2z 2 d 2 , sin(k If we take both cos(k2z d2 ), sin(k3z d3 ) up to the second order, that is cos(k1,2z d1,2 ) = 1 − 12 k1,2z 1,2z d1,2 ) = k1,2z d1,2 . 1,2 2
t ω − After some simple derivation, we can get q 2 = µteff εeff c2
µteff 2 k . µzeff x
t are simultaneously For normal incidence, when µteff and εeff
t have different signs, q 2 is negative, the periodic structure can realize negative refraction. On the other hand, when µteff and εeff negative and the electromagnetic wave cannot propagate in the periodic structure, which indicates a gap. To one’s interest, when |klz |dl is not very large, e.g., |klz |dl < 0.5, the zero–φ gap in Refs. [14,15] just coincides with the gap mentioned above. Thus, the
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boundary positions of the gap can be easily figured out by the results of this Letter. For instance, in Ref. [14], the boundary of the zero–φ gap is calculated to be 4.53557 × 109 and 7.90569 × 109 rad/s which is almost the same as the numerical result, with kd is about 0.4. From this, we can see the usefulness of the EMA method used in this Letter. However, for the large value of |klz |dl , the approximation that sin(klz dl ) ≈ klz dl , and cos(klz dl) ≈ 1 cannot be used. Thus, the EMA method in the present Letter cannot work. Then, to get the effective parameters of the system (bilayer or finite periodic structure), we can use the numerical methods. Here, we show one of these methods and compare it with the method in this paper to further analyze the EMA method. Consider the normal incident case, the total transfer matrix is obtained from the product of all the matrices of the layers. m11 m12 , Ttotal = (M1 M2 )N = (14) m21 m22 in which, N is the number of periods in the periodic structure, and N = 1 refers to the bilayer. To compare Eq. (14) with the first expression in Eq. (7), we can get the effective parameters of the system, z 2 z m12 keff c c arccos(m11 ) 1 keff z t keff (15) = = t . , µteff = − , εeff d ω ω µeff z and µzeff cannot be figured out in the method. Moreover, for The details of the method is dwelt on in Ref. [15]. However, εeff the oblique incidence, i.e., sin2 (θ ) = 0, the effective permittivity and permeability tensors of the system cannot be calculated. One can only give out the effective permittivity and effective permeability, which must be θ -dependent [16]. Actually, for large |klz |dl , the effective permittivity and permeability εeff (µeff ) are not only dependent on ε1 , ε2 (µ1 , µ2 ), εeff (µeff ) is also a function (t,z) (t,z) of µ1 , µ2 (ε1 , ε2 ). Only when the condition |klz |dl 1, the coupling of ε and µ vanishes, then εeff and µeff have the simple and analytic expressions as Eqs. (8) and (9).
3. Amplification of evanescent wave It is well known that negatively refractive materials can amplify evanescent wave due to the excitation of the surface polaritons at the interface between negatively refractive material and positively refractive material [17]. This is one of the most important properties of the negatively refractive materials. To confirm further the equivalence between negatively refractive material and the bilayer made of negative permittivity layer and negative permeability layer, we shall investigate the surface polaritons supported by the bilayer and calculate the transmission coefficient of the evanescent wave. The transmission coefficient [the absolute value of A4 /A1 in Eq. (1)] of evanescent wave can be easily obtained from the transfer matrix method. Using the method in Refs. [12,18,19] and the transfer matrix, we can obtain the dispersion relation of surface polariton supported by this bilayer [20], f (ω, kx ) = cosh(α2 d2 ) cosh(α3 d3 ) + + where α 1 = α4 =
α 2 µ2 + α22 µ23 α22 µ21 + α12 µ22 sinh(α2 d2 ) cosh(α3 d3 ) + 3 2 sinh(α2 d2 ) sinh(α3 d3 ) 2α2 µ1 α1 µ2 2α3 µ2 α2 µ3
α42 µ23 + α32 µ24 cosh(α2 d2 ) cosh(α3 d3 ) = 0, 2α4 µ3 α3 µ4
(16)
ω2 kx2 − 2 εs µs , c
α2 =
ω2 kx2 − 2 ε2 µ2 , c
α3 =
kx2 −
ω2 ε3 µ3 , c2
(17)
and here kx can be larger than k0 . In Fig. 3(a), we plot f (ω, kx ) as a function of frequency for kx /k0 = 2.5. It can be shown that there exist two points of intersection between f (ω, kx ) and the x-axis. This indicates that there are two angular frequencies that satisfy Eq. (16). In other words, for kx /k0 = 2.5, at these two angular frequencies, the surface polaritons can be excited by the bilayer. The transmission coefficient |A4 /A1 | as a function of angular frequency for kx /k0 = 2.5 is shown in Fig. 3(b). Comparing carefully Fig. 3(a) with Fig. 3(b), we can find that the positions of the two points of intersection in Fig. 3(a) are just the positions at which the transmission coefficient diverges. Hence, the excitation of surface polariton can greatly contribute to the amplification of the evanescent wave. And we can see that in the effectively negative refraction region, the bilayer can amplify the evanescent wave. To see the details clearly, we plot the distribution of the electric field of an evanescent wave in the bilayer in Fig. 3(c) near the intersection angular frequency in the negative refraction region. At the first interface between air and the layer 2 (negative permeability), there is an anti-surface polariton, and at the second interface between layer 2 (negative permeability) and layer 3 (negative permittivity), a surface polariton is excited, where the evanescent wave reaches the peak and then falls. Since the layer 3 and air have both positive permeability (in this case, µ3 = 1), for the TE mode wave, the signs of the slope of the electric field at both sides of the third interface are the same, so that no (anti-)surface polariton can be excited. Therefore, we can see that the amplification of the evanescent wave from the bilayer is a total effect of the excitation of the (anti-)surface polaritons at the first and
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(a)
(b)
(c) Fig. 3. The transmission coefficient and f (ω, kx ) as a function of angular frequency are plotted in (a) and (b). The electric field distribution in the bilayer is plotted in (c), for ω = 4.03 × 109 rad/s.
second interfaces, for TE mode wave. For the TM mode wave, the excitation of surface polaritons at the second and third interfaces will contribute to the amplification of the evanescent wave of the bilayer.
4. Summary
In conclusion, using EMA method, we have investigated the equivalence between negatively refractive material and the bilayer made of one negative permittivity layer and one negative permeability layer. We first derive the analytic expressions of all six parameters of the effective permittivity and permeability, under the condition that |klz |dl 1 is satisfied. It is found that the bilayer can be effectively regarded as a uniaxial material and the bilayer can realize negative refraction at the frequency region of [4 × 109 rad/s, 4.714 × 109 rad/s]. And we also analyze the refraction properties and the cutoff condition of the bilayer as well. With the aid of the EMA method, we further investigate the periodic structure when |klz |dl is small, and the case when |klz |dl is large is investigated, too. To confirm the equivalence, we investigate the surface polaritons excited at the interfaces of the bilayer, and find that the bilayer can amplify the evanescent wave indeed as well as the negatively refractive materials.
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Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10204017 and the Natural Science of Jiangsu Province under Grant No. BK2002038. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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