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

Optics Express, Vol. 14, Issue 3, pp. 1161-1166 (2006)

http://dx.doi.org/10.1364/OE.14.001161

Acrobat PDF (617 KB)

### 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

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]

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]

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]

## 2. Electromagnetic fields inside and outside the slab

*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

*e*-

_{1}*e*-

_{2}*e*coordinate system, where

_{3}*μ*,

_{i}*ε*are along with the principle axes

_{i}*e*(

_{i}*i*=1,2,3), respectively, and the

*e*-

_{1}*e*-

_{2}*e*, coordinate system can be obtained by anti-clockwise rotating the

_{3}*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:

*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

*α*=

*μ*

_{1}sin

^{2}

*θ*+

*μ*

_{3}cos

^{2}

*θ*,

*β*= -2sin

*θ*cos

*θ*(

*μ*

_{1}-

*μ*

_{3})and

*γ*=

*μ*

_{1}cos

^{2}

*θ*+

*μ*

_{3}sin

^{2}

*θ*.

*φ*, where

*k*,

_{z1}*k*and

_{z2}*k*satisfy the dispersion relation of Eq. (3);

_{x}*a*,

*b*,

*c*and

*d*are the elements of the inversed tensor of

*μ*̿, and

*a*=

*α*/

*μ*

_{1}

*μ*

_{3},

*b*=

*c*= -

*β*/2

*μ*

_{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.

*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:

*p*

_{1}=(-

*ak*

_{z1}+

*bk*+

_{x}*k*/

_{tz}*μ*

_{0})exp(

*ik*

_{z1}

*z*

_{0}) ,

*p*

_{2}= (-

*ak*

_{z2}+

*bk*+

_{x}*k*/

_{tz}*μ*

_{0})exp(

*ik*

_{z2}

*z*

_{0}) ,

*P*

_{3}= -

*ak*

_{z1}+

*bk*-

_{x}*k*/

_{iz}*μ*

_{0}, and

*P*

_{4}= -

*ak*

_{z2}+

*bk*-

_{x}*k*/

_{iz}*μ*

_{0}.

## 3. Lateral displacement of the Gaussian beam

*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

*k*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

_{z}*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.,

*S*̅

_{1}has a different direction of the tangential component compared to the incident beam’s, while the dashed one will yield a positive shift, or

*S*̅

_{2}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

*ω*= 2

_{ij}*πf*,

_{ij}*i*=

*ε*,

*μ*and

*j*=

*o*,

*p*. Assume

*f*= 0GHz ,

_{εo}*f*= 12.25GHz,

_{εp}*f*= 10.6GHz,

_{μo}*f*= 12.2GHz ,

_{μp}*γ*=

_{ε}*γ*= 0.001,

_{μ}*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.

## 4. Experimental verification

*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]

*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.

## 5. Conclusion

*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

## References and links

01. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

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. |

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. |

04. | D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. |

05. | D. R. Smith, P. Kolinko, and D. Schurig, “Negative Refraction in Indefinite Media,” J. Opt. Soc. Am. B |

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. |

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 |

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 |

**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

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### References

- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292,77-79 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90,077405 (2003). [CrossRef] [PubMed]
- D. R. Smith, P. Kolinko, D. Schurig, "Negative Refraction in Indefinite Media," J. Opt. Soc. Am. B 21, 1032-1043 (2004). [CrossRef]
- 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]
- 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]
- 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]

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