OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 3 — Feb. 6, 2006
  • pp: 1161–1166
« Show journal navigation

Lateral displacement of a Gaussian beam transmitted through a one-dimensional left-handed meta-material slab

Yu Zhong, Lixin Ran, Xiangxiang Cheng, and Jin Au Kong  »View Author Affiliations


Optics Express, Vol. 14, Issue 3, pp. 1161-1166 (2006)
http://dx.doi.org/10.1364/OE.14.001161


View Full Text Article

Acrobat PDF (617 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In this paper, the transmission of a Gaussian beam passing through a slab made of a one-dimensional left-handed meta-material (1D LHM) is studied. The analytical solution of the electric and the magnetic fields inside and outside the slab are given. The calculation of the power flow of the beam predicts that in the negative pass band of the 1D LHM, there exist different directions of lateral displacements. Such phenomenon is further verified by experiment.

© 2006 Optical Society of America

1. Introduction

Since the left-handed meta-material (LHM) was experimentally verified in 2001 [1

01. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

], more and more interests in this field were aroused. The lateral displacement of a Gaussian beam transmitted through an isotropic LHM slab has been studied theoretically and experimentally [2

02. J. A. Kong, B.-I. Wu, and Y. Zhang, “A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability,” Microwave Opt. Technol. Lett. 33, 136–139 (2002). [CrossRef]

,3

03. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Beam Shifting Experiment for the Characterization of Left-handed Properties,” J. Appl. Phys. 95, 2238–2241 (2004). [CrossRef]

]. Different from the isotropic LHM or right-handed material (RHM), which are with circular dispersion relation shapes, and the anisotropic RHM, which is with elliptical dispersion relation shape, a one-dimensional (1D) LHM with only one negative principal element in permittivity tensor and one negative principal element in permeability tensor in the orthogonal direction respecting to the negative permittivity element, has a hyperbolic shaped dispersion relation curve [4–6

04. D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]

], which can be fabricated using S-shaped or Ω-shaped split ring resonators [7

07. H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed Materials Composed of Only S-shaped Resonators,” Phys. Rev. E 70, 057605 (2004). [CrossRef]

,8

08. L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, “Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss,” Phys. Rev. B 70, 073102 (2004). [CrossRef]

]. In this paper, the lateral displacement of a Gaussian beam obliquely incident into an 1D LHM slab is studied. The analytical solutions for the electric and the magnetic fields inside and outside the slab are provided, showing that in the negative pass band of the 1D LHM, the lateral displacement can either shifts to the positive or the negative direction, which is different from isotropic LHM case, whose shift can only be negative [2

02. J. A. Kong, B.-I. Wu, and Y. Zhang, “A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability,” Microwave Opt. Technol. Lett. 33, 136–139 (2002). [CrossRef]

,3

03. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Beam Shifting Experiment for the Characterization of Left-handed Properties,” J. Appl. Phys. 95, 2238–2241 (2004). [CrossRef]

]. Further more, experimental result is also provided to verify the analytical solutions by examining the distribution of the time-averaged power flow at the second interface of the slab where the transmitted Gaussian beam emerges. We see that, the experimental result is in accordance with the analytical result.

2. Electromagnetic fields inside and outside the slab

Fig. 1. Illustration of the lateral displacement of an obliquely incident Gaussian beam.

Consider an 1D LHM slab located in a x-y-z coordinate system depicted in Fig. 1, whose first and second boundaries are located at z=0 and z=z 0, respectively. Let a Gaussian beam be incident obliquely onto the first boundary at Point “A”, and line “AB” be the normal of the two boundaries. Assume the 1D LHM is with a permittivity tensor diag[ε 1, ε 2, ε 3] and a permeability tensor diag[μ 1, μ 2, μ 3] in an e1-e2-e3 coordinate system, where μi,εi are along with the principle axes ei(i=1,2,3), respectively, and the e1-e2-e3, coordinate system can be obtained by anti-clockwise rotating the x-y-z system θ angles around y-axis. The permittivity and permeability tensors described in the x-y-z coordinates are then as follows:

ε̿=[ε1cos2θ+ε3sin2θ0sinθcosθ(ε1ε3)0ε20sinθcosθ(ε1ε3)0ε1sin2θ+ε3cos2θ]
(1)
μ̿=[μ1cos2θ+μ3sin2θ0sinθcosθ(μ1μ3)0μ20sinθcosθ(μ1μ3)0μ1sin2θ+μ3cos2θ]
(2)

in which only ε 2 and μ 1 have negative real parts. For a wave linearly E-polarized along y-axis, the dispersion relation of the 1D LHM is

αkz2+βkxkz+γkx2ω2ε2μ1μ3=0,
(3)

where α = μ 1 sin2 θ + μ 3 cos2 θ, β = -2sin θ cos θ(μ 1 - μ 3)and γ = μ 1 cos2 θ + μ 3 sin2 θ.

Assume a Gaussian beam

Eiy=dkxexp[i(kxx+k0zz)]ψ(kx)
(4)

is incident from region 1 into the slab at an incident angle φ, where ψ(kx)=g2πexp{[g2(kxkixc)24]} is the Gaussian spectrum. The total electric and magnetic fields are expressed as follows:

In region 1,

E1y=dkxψ(kx)(eikizz+Reikizz)eikxx
(5.a)
H1x=1ωμ0dkxψ(kx)kiz(eikizz+Reikizz)eikxx
(5.b)
H1z=1ωμ0dkxψ(kx)kx(eikizz+Reikizz)eikxx
(5.c)

In region 2,

E2y=dkxψ(kx)(ξeikz1z+ζeikz2z)eikxx
(6.a)
H2x=1ωdkxψ(kx)[a(ξkz1eikz1z+ζkz2eikz2z)+b(ξeikz1z+ζeikz2z)kx]eikxx
(6.b)
H2z=1ωdkxψ(kx)[c(ξkz1eikz1z+ζkz2eikz2z)+d(ξeikz1z+ζeikz2z)kz]eikxx
(6.c)

in which kz1, kz2 and kx satisfy the dispersion relation of Eq. (3); a, b, c and d are the elements of the inversed tensor of μ̿, and a = α/μ 1 μ 3, b = c = -β/2μ 1 μ 3, d = γ/μ 1 μ 3 ; ξ, ζ are the transmission or reflection coefficients of the beams in the 1D LHM slab.

In region 3,

E3y=dkxψ(kx)Teiktzzeikxx
(7.a)
H3x=1ωμ0dkxψ(kx)Tktzeiktzzeikxx
(7.b)
H3z=1ωμ0dkxψ(kx)Tkxeiktzzeikxx
(7.c)

The coefficients R, ξ, ζ, and T can be given by matching the boundary conditions for tangential electric and magnetic fields at z=0 and z=z 0, respectively:

ξ=2p2kiz(p1p4p2p3)μ0
(8)
ζ=2p1kiz(p1p4p2p3)μ0
(9)
R=2(p2p1)kiz(p1p4p2p3)μ01
(10)
T=2(p2ei(kz1ktz)z0p1ei(kz2ktz)z0)kiz(p1p4p2p3)μ0
(11)

where p 1=(-ak z1 + bkx + ktz / μ 0)exp(ik z1 z 0) , p 2 = (-ak z2 + bkx + ktz / μ 0)exp(ik z2 z 0) , P 3 = -ak z1 + bkx - kiz / μ 0 , and P 4 = -ak z2 + bkx - kiz / μ 0.

Finally the time-averaged normal and tangential power flow (Poynting vector) can be calculated by

S¯nz=12Re{Eny·Hnx*}
(12)
S¯nx=12Re{Eny·Hnz*}
(13)

where the subscript n = 1, 2, 3 denotes different regions.

3. Lateral displacement of the Gaussian beam

Fig. 2. K-surfaces for the illustration of the lateral shifts for a Gaussian beam incidence.

Here we define: if the directions of the tangential power flow in and out of the slab are the same, the lateral shift of the beam is positive, and vise versa. In Fig. 1, a positive shift means the outgoing beam at the second boundary locates in the upper side to the normal line “AB”, and a negative shift means the opposite situation. Notice that for an RHM slab, the shift must be positive, and for an isotropic LHM, the shift must be negative. From Eq. (3), when the real part of α changes from negative to positive with frequency, the solution for kz changes its sign too, which leads to the change of the direction of the tangential power flow in the 1D LHM, concluded from Eq. (6.c) and Eq. (13). The above discussion could be illustrated by Fig. 2, in which the solid hyperbola represents the k-surface of the 1D LHM when the real part of α is negative, and the dashed one represents the case when the real part of α is positive. Knowing that the Poynting vector is always normal to the k-surface, from Fig. 2, we can see that the solid hyperbola will yield a negative lateral shift, i.e., S̅1 has a different direction of the tangential component compared to the incident beam’s, while the dashed one will yield a positive shift, or S̅2 has the same direction of the tangential component as the incident beam’s. When the Lorentz model is applied to μ 1 and ε 2, say

ε2=1ωεp2ωε02ω2ωε02+iγεω
(14)
μ1=1ωμp2ωμo2ω2ωμo2+iγμω
(15)

where ωij = 2πfij, i = ε,μ and j = o,p. Assume fεo = 0GHz , fεp = 12.25GHz, fμo = 10.6GHz, fμp = 12.2GHz , γε = γμ = 0.001, ε 1 = ε 3 = ε 0, and μ 2 = μ 3 = μ 0. For θ = 18.4°, we see that the real part of α goes to zero when f = 10.78 GHz, indicating that in the first region of the negative pass band, say 10.6 GHz < f < 10.78 GHz (Re{α} < 0), the direction of the tangential power flow in the 1D LHM slab is toward negative direction; and in the second region, say 10.78 GHz < f < 12.2 GHz (Re{α} > 0), the direction of the tangential power flow is positive. For θ = 45°, the separating frequency changes to 11.43GHz, and the negative and positive shift regions change to 10.6 GHz < f < 11.43 GHz and 11.43 GHz < f < 12.2 GHz, respectively. All above can be illustrated in a 2D form by calculating the analytical result of the normal power flow demonstrated by Eq. (12) at the second interface of the slab when the Gaussian beam is incident from air onto the 1D LHM slab at point “A” with an incident angle φ = 45° and g = 0.2 (the Gaussian spectrum parameter defined in Eq. (4)), as shown in Fig. 1. The results are presented in Fig. 3, where the vertical axes represent the location of the transmitted beam, and the originals 0 correspond to the point “B” in Fig. 1. From Fig. 3, we see that in the lower frequency regions, from 10.6 GHz to 10.78 GHz for θ = 18.4° and from 10.6 GHz to 11.43 GHz for θ = 45°, the beam has negative shifts, while in the higher frequency regions, from 10.78 GHz to 12.2 GHz for θ = 18.4° and from 11.43 GHz to 12.2 GHz for θ = 45°, positive shifts appear.

Fig. 3. The normalized power flow calculated at the second interface of the 1D LHM slab. (a)θ = 18.4°, (b)θ = 45°

4. Experimental verification

To verify the above analytical conclusion, the S-shaped 1D LHM sample with a negative pass band occupied from 10 to 12 GHz is chosen to construct the slab with a rotated angle θ = 18.4° [7

07. H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed Materials Composed of Only S-shaped Resonators,” Phys. Rev. E 70, 057605 (2004). [CrossRef]

]. The schematic of the experimental setup is depicted in Fig. 4, in which the point “A” and “B” correspond to the same points in Fig. 1. For the experimental convenience to detect the transmission power, the detector is moved along axis- X'. Although axis- X' is not parallel to the second interface of the 1D LHM slab, the original point 0 still corresponds to the point “B”, so that we can use the axis- X' to define the positive shift region and negative shift region.

Fig. 4. Schematic of the experimental setup for detecting the lateral shift of the incident Gaussian beam.

Fig. 5. Normalized transmission power detected at the second interface of the S-shaped 1DLHM slab.

5. Conclusion

Since the 1D LHM studied in this paper has a unique hyperbola dispersion relation, analytically we conclude that in the negative pass band, a rotated 1D LHM slab will bring both negative and positive lateral shifts for an obliquely incident Gaussian beam. This is also verified by the experimental result demonstrated above. The critical frequency, who separates the negative pass band into two parts, is decided by the rotating angle θ and the dispersion property of the negative permeability element μ 1, if the incident beam is E-polarized. According to the duality, we can also draw a similar conclusion for a H-polarized beam. Such structure can be used as a spatial beam splitter or diplexer, by giving an appropriate dispersion property and rotated angle of the material.

Acknowledgments

This work is supported by Chinese Natural Science Foundation under contract 60531020, 60371010 and 60201001.

References and links

01.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

02.

J. A. Kong, B.-I. Wu, and Y. Zhang, “A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability,” Microwave Opt. Technol. Lett. 33, 136–139 (2002). [CrossRef]

03.

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Beam Shifting Experiment for the Characterization of Left-handed Properties,” J. Appl. Phys. 95, 2238–2241 (2004). [CrossRef]

04.

D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]

05.

D. R. Smith, P. Kolinko, and D. Schurig, “Negative Refraction in Indefinite Media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004). [CrossRef]

06.

T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, “Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials”, IEEE Trans. Microw. Theory and Technol. 53, 1443–1450 (2005). [CrossRef]

07.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed Materials Composed of Only S-shaped Resonators,” Phys. Rev. E 70, 057605 (2004). [CrossRef]

08.

L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, “Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss,” Phys. Rev. B 70, 073102 (2004). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(230.1360) Optical devices : Beam splitters
(350.4010) Other areas of optics : Microwaves

ToC Category:
Metamaterials

History
Original Manuscript: November 28, 2005
Revised Manuscript: January 30, 2006
Manuscript Accepted: January 30, 2006
Published: February 6, 2006

Citation
Yu Zhong, Lixin Ran, Xiangxiang Cheng, and Jin Au Kong, "Lateral displacement of a Gaussian beam transmitted through a one-dimensional left-handed meta-material slab," Opt. Express 14, 1161-1166 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1161


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001). [CrossRef] [PubMed]
  2. J. A. Kong, B.-I. Wu, and Y. Zhang, "A Unique Lateral Displacement of a Gaussian Beam Transmitted through a Slab with a Negative Permittivity and Permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002). [CrossRef]
  3. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Beam Shifting Experiment for the Characterization of Left-handed Properties," J. Appl. Phys. 95,2238- 2241 (2004). [CrossRef]
  4. D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003). [CrossRef] [PubMed]
  5. D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004). [CrossRef]
  6. T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong, "Refraction Laws for Anisotropic Media and Their Application to Left-handed Metamaterials", IEEE Trans. Microw. Theory and Technol. 53,1443-1450 (2005). [CrossRef]
  7. H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Left-handed Materials Composed of Only S-shaped Resonators," Phys. Rev. E 70,057605 (2004). [CrossRef]
  8. L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave Solid-state Left-handed Material with a Broad Bandwidth and an Ultralow Loss," Phys. Rev. B 70,073102 (2004). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited