## Negative refractions and backward waves in biaxially anisotropic chiral media

Optics Express, Vol. 14, Issue 13, pp. 6322-6332 (2006)

http://dx.doi.org/10.1364/OE.14.006322

Acrobat PDF (132 KB)

### Abstract

We investigate the refraction and propagation properties of electromagnetic waves which are incident from free space to a biaxially anisotropic chiral medium. Such a chiral medium can be realized by putting chiral elements pointing to two directions into a host medium. When the host medium is a normally isotropic dielectric, no negative refraction and/or backward waves are supported for the propagating eigenwaves in the chiral medium. When the host medium changes to an anisotropic dielectric or an electric plasma, however, negative refractions and backward waves can be realized separately or even simultaneously if we choose the medium parameters properly. Numerical simulations validate our theoretical analysis.

© 2006 Optical Society of America

## 1. Introduction

2. J. B. Pendry, “A chiral route to negative refraction,” Science **306**, 1353 (2004). [CrossRef] [PubMed]

*et al.*have analyzed the possibility on existence of negative refraction in chiral nihility [3

3. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. **17**, 695 (2003). [CrossRef]

4. T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E **69**, 026602 (2004). [CrossRef]

5. Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express **13**, 4974 (2005). [CrossRef] [PubMed]

6. S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. **3**, 107 (2005). [CrossRef]

7. C. Monzon and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett. **95**, 123904 (2005). [CrossRef] [PubMed]

8. Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B **73**, 113104 (2006). [CrossRef]

8. Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B **73**, 113104 (2006). [CrossRef]

9. T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a material with simultaneous mirror-conjugated and racemic chirality characteristics,” New J. Phys. **7**, 165 (2005). [CrossRef]

9. T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a material with simultaneous mirror-conjugated and racemic chirality characteristics,” New J. Phys. **7**, 165 (2005). [CrossRef]

## 2. General Formulations

*I̿*

_{t}=

*x̂*

*x̂*+

*yŷŷ*, κ is the chirality factor, and

*ε*

_{0}and

*µ*

_{0}are permittivity and permeability of free space. Clearly, both the permittivity and permeability are uniaxial tensors which are

*ε*

_{t}and

*µ*

_{t}in the

*x*and

*y*directions and

*ε*

_{z}and

*µ*

_{z}in the

*z*direction, while the chirality appears in

*y*and

*z*directions. Hence the overall medium behaves a biaxial property. Such a kind of chiral medium can be easily constructed by embedding chiral objects like wire helices or M

*ö*bius strips regularly in two directions in the host dielectric.

*θ*

_{i}. Assume that the plane of incidence is within the

*yoz*plane so that

*k̄*

_{i}=

*k*

_{y}

*ŷ*+

*k*

_{0z}

*ẑ*and the electric field is written as

10. J. L. Tsalamengas, “Interaction of electromagnetic waves with general bianisotropic slabs,” IEEE Trans. Microwave Theory Tech. **40**, 1870 (1992). [CrossRef]

*p*=

*ω*

^{2}(

*ε*

_{z}

*µ*

_{z}-

*κ*

^{2}

*ε*

_{0}

*µ*

_{0}) and

*zĒ*(

*H̄*)=

*ik*

_{1z}

*Ē*(

*H̄*). From Eq. (3), it is clear that the following relation always holds

*M̿*. From Eqs. (5) and (6), the dispersion relations for eigenwaves within the biaxially chiral medium can be obtained after some lengthy algebra

*q*=

*ε*

_{t}

*µ*

_{t}+

*ε*

_{z}

*µ*

_{z}-

*ε*

_{z}

*µ*

_{t}-

*ε*

_{t}

*µ*

_{z}. When the chirality disappears, i.e.,

*κ*=0, Eq. (7) obviously turns into the following form

3. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. **17**, 695 (2003). [CrossRef]

5. Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express **13**, 4974 (2005). [CrossRef] [PubMed]

*A*is the amplitude of vectors, and

*S*

_{y}and

*S*

_{1z}in the chiral medium as

*µ*

_{t}=

*µ*

_{z}=

*µ*

_{0}, the eigenvectors of electric and magnetic fields are simplified as

*y*and

*z*directions can be obtained in simple forms as

*z*direction to satisfy the radiation condition. On the other hand, the transverse components of incident and refracted wave vectors must keep identical according to the boundary condition and the phase matching. When

*S*

_{y}has the same sign with

*k*

_{y}, a positive refraction occurs at the medium interface. On the contrary, an opposite sign of

*k*

_{y}and

*S*

_{y}implies that a negative refraction is supported on the boundary. Similarly, when

*S*

_{1z}has the same sign with

*k*

_{1z}, the eigenwaves within the biaxially chiral medium are propagating away from the medium interface. That is to say, they are travelling forwardly due to the positive longitudinal wavenumber

*k*

_{1z}. When they have the opposite signs, the eigenwaves will then travel toward the medium interface, and they are actually backward waves.

*κ*>0 in the following discussions. For the case when

*κ*<0, similar conclusions may be drawn as we can see later.

## 3. Theoretical Analysis and Numerical Simulations

*ε*

_{t}=

*ε*

_{z}>0 and

*µ*

_{t}=

*µ*

_{z}=

*µ*

_{0}. The corresponding dispersion equation is easily obtained from Eqs. (7)–(13) as

*p*

^{+}and

*p*

^{-}waves. Clearly, when the chirality satisfies the condition

*p*

^{+}and

*p*

^{-}waves exist in the chiral medium with a circular dispersion relation. When

*p*

^{-}wave becomes evanescent and cannot be supported by the chiral medium. Generally, the corresponding

*y*and

*z*components of the Poynting vector in the chiral half space are expressed as

*p*

^{+}wave,

*S*

_{y}and

*S*

_{1z}always have the same sign with

*k*

_{y}and

*k*

_{1z}for all possible κ, indicating that neither negative refraction nor backward wave exists in such a chiral medium. For the

*p*

^{-}wave,

*S*

_{y}and

*S*

_{1z}are still parallel with

*k*

_{y}and

*k*

_{1z}, respectively, when

*p*

^{-}wave can be refracted negatively at the interface. However it becomes evanescent within the chiral medium, as we have mentioned earlier.

*S̄*and wave vector

*k̄*for incident and refracted waves have been plotted with

^{-1}

*S*

_{1z}/

*S*

_{y}=tan

^{-1}

*k*

_{1z}/

*k*

_{y}. As a consequence, we conclude that neither negative refraction nor backward wave exists in the propagating eigenwaves if the host medium is the nonmagnetic isotropic dielectric.

*ε*

_{t}≠

*ε*

_{z}) but it still keeps nonmagnetic (

*µ*

_{t}=

*µ*

_{z}=

*µ*

_{0}), the dispersion equation becomes much more complicated. From Eq. (9),

*f*may be imaginary since

*ap*

^{2}+

*bp*+

*c*may be less than zero with the change of incident angles. Therefore the transverse wavenumbers of the eigenwaves in the biaxially chiral medium may be complex, implying that such a wave would decay exponentially when propagating in the biaxially chiral medium. Moreover, the right-hand side of Eq. (7) may also be negative even when

*f*>0 is satisfied, and the wave becomes totally evanescent.

*S*

_{y}and

*S*

_{1z}may become complex from Eqs. (30) and (31), whose real part represents the electromagnetic power going away from the interface and imaginary part stands for the interchange of electric and magnetic energies. Since the transverse wavenumber

*k*

_{y}is continuous at the interface between free space and the chiral medium and has a positive real value as we have assumed in Fig. 1, the refraction property is only dependent on the sign of Re(

*γ*) based on Eq. (30). When Re(

*γ*)>0, the power flow

*S*

_{y}has the same direction as the transverse wavenumber

*k*

_{y}, which shows positive refraction at the medium interface and vice versa. Based on Eq. (31), however, the propagation properties of eigenwaves in the biaxially chiral medium are determined by Re(

*τ*)·L(

*k*

_{1z}). Here, L is a function of

*k*

_{1z}which keeps zero if the real part of

*k*

_{1z}is zero and keeps unity otherwise. The reason we introduce the function L is because

*k*

_{1z}may become pure imaginary as we have mentioned earlier. It is meaningless to discuss the backward propagation for evanescent waves.

*γ*) and Re(τ)·L(

*k*

_{1z}) with respect to incident angles

*θ*

_{i}numerically. For simplicity, we define a new function as

*α*(

*γ*) and

*α*(τ)·L(

*k*

_{1z}) for different incident angles.

*ε*

_{t}=

*ε*

_{z}=2

*ε*

_{0}and

*κ*=0.2, the corresponding curves of

*α*(

*γ*) and

*α*(τ)·L(

*k*

_{1z}) are shown in Figs. 3(a) and 3(b). Obviously, both

*p*

^{+}and

*p*

^{-}waves are positively refracted at the interface and propagating forwardly in the biaxially chiral medium. When the chirality increases to

*κ*=1.5, the

*α*(

*γ*) and

*α*(τ) ·L(

*k*

_{1z}) curves are illustrated in Figs. 3(c) and 3(d), where the

*p*

^{+}wave still keeps positive refraction and forward propagation. For the

*p*

^{-}wave, however, it becomes evanescent since the real part of the Poynting vector

*S*

_{1z}turns to be zero although

*α*(

*γ*)<0 in Fig. 3(c). In this case, the

*p*

^{-}wave are totally travelling along the interface. Numerical results are consistent with the theoretical analysis earlier.

*ε*

_{t}=8

*ε*

_{0},

*ε*

_{z}=2

*ε*

_{0}, and κ=0.3, the curves of

*α*(

*γ*) and

*α*(τ)·L(

*k*

_{1z}) are illustrated in Figs. 4(a) and 4(b). In such a case, both eigenwaves have positive refraction and forward propagation like that in Figs. 3(a) and 3(b). However, when

*κ*becomes larger (

*κ*=1.5), negative refractions may occur on the medium interface for both

*p*

^{+}and

*p*

^{-}waves, as illustrated in Fig. 4(c). From Fig. 4(c), we also notice that the

*p*

^{-}wave changes to positive refraction as the incident angle

*θ*

_{i}is greater than 70°. The two eigenwaves are still forward waves under such a case, as shown in Fig. 4(d).

*α*(

*γ*) and

*α*(τ)·L(

*k*

_{1z}) at different incident angles, as demonstrated in Fig. 5. When

*ε*

_{t}=-2

*ε*

_{0},

*ε*

_{z}=

*ε*

_{0}, and

*κ*=0.3, the value of

*α*(

*γ*) is always equal to one for both

*p*

^{+}and

*p*

^{-}waves. That is to say, both eigenwaves are positively refracted at the medium interface, as shown in Fig. 5(a). Furthermore, from Fig. 5(b), the

*p*

^{+}wave are backward wave when

*θ*

_{i}<35° or

*θ*

_{i}>75°. In the middle region of 35°<

*θ*

_{i}<75°, the

*p*

^{+}wave is totally evanescent since the power flow along the

*z*direction is always zero. For the

*p*

^{-}wave, it is backward wave when the incident angle is less than 35°, and becomes evanescent for larger incident angles.

*κ*increases to 1.5, both eigenwaves are refracted negatively as we can see from Fig. 5(c). These eigenwaves travel in the biaxially chiral medium backwardly under when the incident angle is less than 57° and become evanescent at larger incident angles, as shown in Fig. 5(d). We remark that the negative refraction and backward wave are supported simultaneously in such a case, which helps to realize the superlens effect.

*z*direction, e.g.

*ε*

_{t}=2

*ε*

_{0},

*ε*

_{z}=-

*ε*

_{0}, and

*κ*=0.3, the

*p*

^{+}wave experiences negative refraction and forward propagation, while the

*p*

^{-}wave has negative refraction only when

*θ*

_{i}<23° and keeps forward propagation, as illustrated in Figs. 6(a) and 6(b). When

*κ*increases to 0.8, only the refraction properties of the two eigenwaves vary with the change of incident angles (see Fig. 6(c)), while both eigenwaves keep forward propagation (see Fig. 6(d)).

*ε*

_{t}≠

*ε*

_{z},

*µ*

_{t}≠

*µ*

_{z}. The propagation and refraction properties of eigenwaves in the chiral medium have been illustrated in Fig. 7, where we choose

*ε*

_{t}=2

*ε*

_{0},

*ε*

_{z}=-

*ε*

_{0},

*µ*

_{t}=2

*µ*

_{0},

*µ*

_{z}=

*µ*

_{0},

*κ*=0.3 and

*κ*=0.8, respectively. From Figs. 7(a) and 7(c), it is clear that there exist transition angles for both eigenwaves to turn from positive (negative) refraction to negative (positive) refraction. But they remain forward propagation in the chiral half space, as illustrated in Figs. 7(a) and 7(d). In comparison to Fig. 6, we find that the magnetic anisotropy mainly affects the refraction behaviors of the eigenwaves.

## 4. Conclusions

## Acknowledgments

## References and links

1. | I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, |

2. | J. B. Pendry, “A chiral route to negative refraction,” Science |

3. | S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. |

4. | T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E |

5. | Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express |

6. | S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. |

7. | C. Monzon and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett. |

8. | Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B |

9. | T. G. Mackay and A. Lakhtakia, “Negative phase velocity in a material with simultaneous mirror-conjugated and racemic chirality characteristics,” New J. Phys. |

10. | J. L. Tsalamengas, “Interaction of electromagnetic waves with general bianisotropic slabs,” IEEE Trans. Microwave Theory Tech. |

**OCIS Codes**

(120.5710) Instrumentation, measurement, and metrology : Refraction

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 10, 2006

Revised Manuscript: May 13, 2006

Manuscript Accepted: May 24, 2006

Published: June 26, 2006

**Citation**

Qiang Cheng and Tie Jun Cui, "Negative refractions and backward waves in biaxially anisotropic chiral media," Opt. Express **14**, 6322-6332 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6322

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### References

- I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic waves in Chiral and Bi-Isotropic Media (Artech House, Boston, 1994).
- J. B. Pendry, "A chiral route to negative refraction," Science 306, 1353 (2004). [CrossRef] [PubMed]
- S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, "Waves and energy in chiral nihility," J. Electromagn. Waves Appl. 17, 695 (2003). [CrossRef]
- T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004). [CrossRef]
- Y. Jin and S. He, "Focusing by a slab of chiral medium," Opt. Express 13, 4974 (2005). [CrossRef] [PubMed]
- S. Tretyakov, A. Sihvola, and L. Jylha, "Backward-wave regime and negative refraction in chiral composites," Photonics Nanostruct. Fundam. Appl. 3, 107 (2005). [CrossRef]
- C. Monzon and D. W. Forester, "Negative refraction and focusing of circularly polarized waves in optically active media," Phys. Rev. Lett. 95, 123904 (2005). [CrossRef] [PubMed]
- Q. Cheng and T. J. Cui, "Negative refractions in uniaxially anisotropic chiral media," Phys. Rev. B 73, 113104 (2006). [CrossRef]
- T. G. Mackay and A. Lakhtakia, "Negative phase velocity in a material with simultaneous mirror-conjugated and racemic chirality characteristics," New J. Phys. 7, 165 (2005). [CrossRef]
- J. L. Tsalamengas, "Interaction of electromagnetic waves with general bianisotropic slabs," IEEE Trans. Microwave Theory Tech. 40, 1870 (1992). [CrossRef]

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