Abrupt change of reflectivity from the strongly anisotropic metamaterial

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Optics Communications 281 (2008) 1941–1944 www.elsevier.com/locate/optcom

Abrupt change of reflectivity from the strongly anisotropic metamaterial Guoan Zheng * Department of Optical Engineering, Zhejiang Univeristy, HangZhou, ZheJiang 310027, China Electromagnetics Academy at Zhejiang University, HangZhou, ZheJiang 310058, China Received 29 July 2007; received in revised form 6 December 2007; accepted 7 December 2007

Abstract The wave reflection from a non-magnetic anisotropic metamaterial, whose principal elements of the permittivity tensor have different signs, is investigated in this paper. It is found that, if the orientation of the optical axis is properly chosen, an extremely small change of the transverse wave number will lead to a dramatically change of the reflectivity at the glancing incidence. The physical insight for this abrupt change of reflectivity is also given by the analysis of the imaginary part of the k-surface. Since the metamaterial discussed here have been experimental realized from GHz to optical frequencies, the proposed abrupt change property of reflectivity may find some potential applications in various calibration devices, because of its extremely sensitivity to the transverse wave number. Ó 2007 Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 41.20.Jb; 42.25.Bs Keywords: Wave reflection; Anisotropic metamaterial; k-Surface

1. Introduction The anisotropy of dielectric media is widely used in the design of various optical devices such as retarding plates, Soleil–Babinet compensator, birefringent prisms and etc. [1]. The performance of these polarization-sensitive applications can be related to the relative difference of the dielectric constants along different directions. However, in the majority of natural anisotropic crystals, the relative difference of the dielectric constants is below 30% [2], and thus it may set up a limitation for further applications of the anisotropy of media. In the passing years, a new class of material named metamaterial [3,4], has draw much attentions both from science and engineering society. Metamaterial owes its properties to the subwavelength resonance of the structure rather than their chemical composition [5], *

Present address: California Institute of Technology, MC 136-93, Pasadena, CA 91125, USA. E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.12.048

and therefore, it is possible to design materials that have properties difficult or impossible to find in nature [6]. With the progress of metamaterial, it is now conceivable that a strongly non-magnetic anisotropic metamaterial can be constructed with its dielectric constants along different directions having different signs. The salient feature of such strongly anisotropic metamaterial is the hyperbola-like ksurface [7], and thus it provides some novel propagation characteristics such as negative refraction, near-field focusing, high impedance surface reflection and frequency selective total oblique transmission, which have been extensively studied in the past (see, for example Refs. [7–10]). Recently, it is proposed that such strongly anisotropic metamaterial can find some applications in far-field sub-diffraction imaging [11,12], and it has been experimental realized in optical region by a composite of altering layers of metal and dielectric [13]. A comparatively lowloss strongly anisotropic metamaterial, which is fabricated by altering semiconductor layers has also been reported recently [14]. In GHz and THz frequencies region, such metamaterial can also

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G. Zheng / Optics Communications 281 (2008) 1941–1944

be realized in a composite of periodically arranged metallic thin wires aligned along the optical axis [15–17]. In this paper, the wave reflection from such non-magnetic strongly anisotropic metamaterial is investigated. It is shown that, if the orientation of the optical axis is properly chosen, an extremely small change of the transverse wave number will lead to a dramatically change of the reflectivity at the glancing incidence. The physical insight for this abrupt change effect is also given by the analysis of the imaginary part of the k-surface. The proposed abrupt change property may find some potential applications in various calibration devices, because of its extremely sensitivity to the transverse wave number.

B e¼@

¼

0

0  sin h cos hðe?  ek Þ

e? 0

 sin h cos hðe?  ek Þ

1

C 0 A; 2 2 e? sin h þ ek cos h

ði ¼ 1; 3Þ

2

2

cos h þ ek sin hÞ þ

k 22z ðek

ð3aÞ 2

2

cos h þ e? sin hÞ 2

þ 2 sin h cos hk x k 2z ðek  e? Þ  ek e? ðx=cÞ ¼ 0;

ð3bÞ

where k0 is the wave number in the air. One should note that there are two solution for k2z based on Eq. (3b), one is k i2z and the other is k r2z . The electric field in different region can be expressed as !

r  Hi Ei ¼ e  ; ix !

¼1

ði ¼ 1; 2; 3Þ:

ð4Þ

The reflection coefficient R can be solved by the boundary condition at z = 0 and z = d, with the result R¼

We consider a non-magnetic uniaxial media with a scalar permeability l = 1 and a permittivity tensor [18] e? cos2 h þ ek sin2 h

k 2x ðe?

i

2. Wave reflection from the anisotropic metamaterial

0

2

k 2x þ k 2iz ¼ k 20 ¼ ðx=cÞ ;

r

ðeik2z d  eik2z d Þ  p  q i

r

eik2z d  s  p  eik2z d  z  q

ð5Þ

;

where p ¼ k 1z  k i2z m þ n; q ¼ ðk 1z  k r2z m þ nÞ, z ¼ k 1z þ k i2z m  n; m¼

s ¼ ðk 1z þ k r2z m  nÞ;

cos2 h sin2 h þ ; e? ek

n ¼ k x sin h  cos h 



 1 1  : e? ek

ð1Þ where h is the angle between the optical axis and the z-axis, as shown in Fig. 1, ek is the dielectric constant along the optical axis, and e\ is the dielectric constant perpendicular to the optical axis. We simply assume d = 2k, ek = 2 + 0.001i and e\ = 2 + 0.01i in our discussion (the general discussion will be given in the next section). Only a TM wave with H-field polarized along y-axis is considered here. The H-field in region 1–3 can be expressed as below !

!

! ik x x

!

!

H2 ¼ y e

ðAe

ik i2z z

H 3 ¼ y T eikx x eik3z z ;

þ Be

ik r2z z

Þ;

z<0

ð2aÞ

06z6d

ð2bÞ ð2cÞ

z > d;

where kx is transverse wave number in x direction, kiz (i = 1, 2, 3) is the wave number along the propagating direction (z-axis) in region 1, 2 and 3. For a given kx, kiz can be solved based on the dispersion relation

x

Region 1

Region 2

Region 3

d

Air

Abrupt change

z

ε⊥

ε II

Air

θ

Fig. 1. A strongly anisotropic metamaterial slab with ek = 2 + 0.001i and e\ = 2 + 0.01i.

b

Magnitude of Reflection Coefficient

!

H 1 ¼ y eikx x ðeik1z z þ Reik1z z Þ;

a

1

θ=30.2 (in degree) Δ k x = 0.001k0

0.8

Region A 0.6

Region B

0.4

Δk x = 0.009 k 0

0.2 0 0.98

0.985

0.99

0.995

1

kx/k0 Fig. 2. (a) The reflection coefficient R is plotted as the function of transverse wave number kx and the orientation of the optical axis h. (b) The abrupt change of reflection coefficient with h = 30.2°.

G. Zheng / Optics Communications 281 (2008) 1941–1944

3. Analysis of the imaginary part of the k-surface The dispersion relation of the strongly anisotropic metamaterial is determined by Eq. (3b), from which one can solve for k2z (k i2z and k r2z ) as below pffiffiffiffi b  D ; ð6Þ k 2z ¼ 2a where a = ek cos2h + e\ sin2h, b = 2 sin h cos hkx (ek  e\), and

Table 1 Conditions and property of four cases of media Media conditions eke\ < 0 eke\ < 0 eke\ > 0 eke\ > 0

Always pass High pass Low pass Always stop

Propagation aeke\ > 0 aeke\ < 0 aeke\ > 0 aeke\ < 0

All kxs kx > kc kx < kc No kxs

θ=30.2 (in degree)

10

Re(k2z)/k0 5

Re(k2z)/k0, Im(k2z)/k0

The reflection coefficient R is plotted as the function of transverse wave number kx and the orientation of the optical axis h in Fig. 2a and the abrupt change of R is shown in Fig. 2b with h = 30.2°. There are two regions in Fig. 2b. In region A, the magnitude of R decreases from 0.9 to 0.1 with the change of transverse wave number Dk x ¼ 0:009k 0 . The abrupt change of R in this region will be explained by the analysis of the imaginary part of k-surface in the next section. In region B, it is easy to understand that when k x ¼ k 0 , the magnitude of R should be 1 (no wave can propagate through the slab), and thus the magnitude of R increases from 0.1 to 1 so rapidly with the change of transverse wave number Dk x ¼ 0:001k 0 .

1943

Im(k2z)/k0

0 1

-5 0.5

-10 -15

0 -0.5

-20 -2

Cutting point ( k c = 0.99 k 0 )

-1 0.9

1

-1

1.1

0

1

2

kx/k0

D ¼ 4ek e? k 2x þ 4aek e? k 20 :

ð7Þ

In our discussion, kx is determined by the incident wave and restricted to be real. Attentions should be given to D in Eq. (7), because it determines the imaginary part of the k-surface. The sign of D can be used to distinguish the nature of the propagating characteristics of the EM wave. D > 0 corresponds to real value of k2z and propagating solutions. D < 0 corresponds to complex value of k2z and exponentially growing or decaying solutions. Four cases of the media are identified based on D, as shown in Fig. 3 and Table 1. Note that the term ‘‘pass” and ‘‘stop” here refers to the transverse wave number kx, not the frequency. The discussion in the following will only focus on the high pass type because the abrupt change behaviour can only be expected in such media. The cutting point kc can be solved based on D = 0, with the result pffiffiffi k c ¼ ak 0 : ð8Þ

Δ

Always pass High pass 2

kx Low pass Cutting point Always stop

Fig. 3. Four cases of media based on the property of D.

Fig. 4. The k-surface of strongly anisotropic media with h = 30.2°, ek = 2 + 0.001i and e\ = 2 + 0.01i.

The idea to achieve the abrupt change of R is to make the cutting point kc very close to k0 by properly choosing the orientation of the optical axis h. The wave with the transverse wave number kc < kx < k0 can pass through the slab while the wave with kx < kc and kx = k0 will be totally reflected from the slab. Therefore, in the extremely short range from kc to k0, the reflectivity will experience an abrupt decrease from 0.9 to 0.1 and then an abrupt increase from 0.1 to 1, as shown in Fig. 2b. Based on Eq. (6), one can see that k i2z and k r2z are complex values for a given kx, and thus the real and the imaginary part should be plotted respectively, as shown in Fig. 4, where the cutting point kc = 0.99k0. It can be concluded from Fig. 4 that the sudden drop of R in region A of Fig. 2b is due to the abrupt change of the imaginary part of the k-surface at the cutting point. The small loss of the media, i.e. the imaginary part of the dielectric constant, has been considered in the discussion above. If the loss become smaller, the phenomena of the abrupt change will be better. We also note that the metamaterial discussed above is intrinsically dispersive and therefore the proposed abrupt change effect can only work for a narrow frequency band. 4. Conclusion In summary, a wave reflection from a non-magnetic strongly anisotropic metamaterial is investigated in this

1944

G. Zheng / Optics Communications 281 (2008) 1941–1944

paper. It is shown that, if the orientation of the optical axis is properly chosen, an extremely small change of the transverse wave number will lead to a dramatically change of the reflectivity at the glancing incidence. The physical explanation for this abrupt change is given by the analysis of the imaginary part of the k-surface. The abrupt change of R from 0.9 to 0.1 and then to 1 can be explained by two facts respectively: the high pass property of the k-surface with the cutting point very close to k0, and the total reflection (jRj = 1) when kx = k0. Since the strongly anisotropic metamaterial discussed here has been experimentally reported from GHz to optical frequencies [13,14,16,17], our result may find some potential applications in the design of various microwave and optical calibration devices. References [1] M. Born, E. Wolf, Principle of Optics, sixth ed., Pergamon, Oxford, 1984. [2] E. Palik (Ed.), The Handbook of Optical Constants of Solids, Academic, London, 1997. [3] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184.

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