## Spatial dispersion and energy in a strong chiral medium

Optics Express, Vol. 15, Issue 8, pp. 5114-5119 (2007)

http://dx.doi.org/10.1364/OE.15.005114

Acrobat PDF (84 KB)

### Abstract

Since the discovery of backward-wave materials, people have tried to realize a strong chiral medium, which is traditionally thought impossible mainly for the reason of energy and spatial dispersion. We compare the two most popular descriptions of a chiral medium. After analyzing several possible reasons for the traditional restriction, we show that a strong chirality parameter leads to positive energy without any frequency-band limitation in the weak spatial dispersion. Moreover, strong chirality does not result in a strong spatial dispersion, which occurs only around the traditional limit point. For strong spatial dispersion where higher-order terms of spatial dispersion need to be considered, the energy conservation is also valid. Finally, we show that realization of strong chirality requires the conjugated type of spatial dispersion.

© 2007 Optical Society of America

## 1. Introduction

## 2. Two major electromagnetic models to describe a chiral medium

5. L. D. Landau and E. M. Lifshitz, *Electromagnetics of continous media, vol. 8 of Course
of Theoretical Physics*, 2nd edition, English,
(Pergamon Press,
1984). [PubMed]

*ε*or

_{DBF}β*μ*can be either positive or negative for two stereoisomer structures. And in this paper, we assume our chiral medium is source free, i.e. no electric or magnetic sources inside the medium. Solving the constitutive relation together with Maxwell’s equations, we can easily get two eigenwaves, which are left and right circularly polarized with different wavevectors.

_{DBF}β9. A. Lakhtakia, *Beltrami Fields In Chiral media*
(World Scientific Publishing Co. Pte. Ltd.,
Singapore, 1994). [CrossRef]

*κ*= 0 and

*χ*≠ 0, it is the Tellegen medium; if

*χ*= 0 and

*κ*≠ 0, as the requirement of reciprocity, it is the Pasteur medium:

*κ*values differentiate two conjugated stereoisomer structures. We assume

*κ*> 0 in the following analysis.

*D*¯ to

*E*¯ and

*B*¯ to

*H*¯. The rotation terms in DBF model include both real and imaginary parts, resulting in a change in the real part and creating the imaginary chiral terms in the Pasteur model, vice versa. In other words, the difference in representations of coupling terms lead to different permittivity and permeability formulations.

## 3. Energy and spatial dispersion in strong chiral medium

12. 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]

*κ*

^{2}>

*με*[7, 12

12. 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]

*κ*

^{2}<

*με*), until we see the fact that artificial Veselago’s medium [13

13. V. G. Veselago, “The electrodynamics of substances
with simultaneously negative values of *ε* and
*μ*,” Sov.
Phys. Usp. **10**, 509 (1968). [CrossRef]

14. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a
negative index of refraction,” Science **292**, 77 (2001). [CrossRef] [PubMed]

12. 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]

*κ*

^{2}>

*με*for the whole frequency range, the energy will still remain positive as long as the permittivity and permeability are positive, under the weak spatial dispersion condition. This is quite different from Veselago’s medium since there is no bandwidth limitation and the frequency dispersive resonances are no longer required. In another words, the strong chiral medium does not contradict energy conservation, at least in the weak spatial dispersion model.

*ε*>0,

*μ*> 0,

*κ*> 0 and

*ε*,

_{DBF}*μ*and

_{DBF}*β*become negative on transformation between Pasteur constitutive relations and DBF relations shown in Eqs. (7)–(9). This was considered absolutely unacceptable before people discovered the possibility of Veselago’s medium. Actually, strong chiral medium can be equivalent to Veselago’s medium for the right circularly polarized wave [12

**17**, 695 (2003). [CrossRef]

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

*ε*and

_{DBF}*μ*have shown such a point. Hence the negative sign in the DBF model is not strange at all, since we realize effective double-negative with a strong chirality parameter instead of simultaneous frequency resonances. For a limiting case, the chiral nihility [12

_{DBF}**17**, 695 (2003). [CrossRef]

*ε*→ 0 and

*μ*→ 0 while

*κ*

*≠*0, the parameters in DBF representation become

*ε*→ ∞,

_{DBF}*μ*→ ∞ and

_{DBF}*β*= -1/(

*ωκ*) remains a finite value after a simple mathematical analysis. There is no evidence that strong chirality cannot exist in this aspect.

*β*, while the strong chirality is represented by the Pasteur model, e.g. the ratio of

*κ*to

*β*and

*ε*/

_{DBF}*ε*or

*μ*/μ versus

_{DBF}*κ*is very close to

*κ*continuously increasing, the spatial dispersion strength falls down very quickly. Therefore, if

*κ*is not near

*κ*is close to

*β*stands for the spatial dispersion of the

_{n}*n*th order. We remark that the above is different from classical nonlinear optics because it is strong spatial dispersion instead of strong field intensity. Hence it is not a power series of

*E*¯ and

*H*¯ fields.

*D*¯ can still be represented as a real part proportional to

*E*¯ and an imaginary part proportional to

*H*¯ with modified coefficients.

*B*¯ has similar representations to

*D*¯. The nonlinear terms contribute to the alteration of effective

*ε*,

*μ*and

*κ*in the Pasteur model, which might be negative, leading to the energy problem again.

## 4. Conclusions

*κ*and

*β*have opposite signs, which necessarily leads to negative

*ε*and

_{DBF}*μ*. Here,

_{DBF}*κ*stands for chirality and β is the coefficient of the first order for spatial dispersion. Strong chirality stems from using one type of spatial dispersion to get the conjugated stereoisomer, or chirality. That is to say, when we manage to use negative

*β*to realize positive

*κ*, or vice versa, there is strong chirality, (seen from Fig. 1). Comparing with the fact that Veselago’s medium requires effective double negativeness from natural double positive materials, getting one type of chirality from the conjugated type of spatial dispersion maybe another substitutive way to support backward-wave propagation.

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

## Acknowledgement

## References and links

1. | L. Pasteur, |

2. | A. J. Fresnel, in |

3. | W. A. Shurcliff and S. S. Ballard, |

4. | Eugene Hecht, |

5. | L. D. Landau and E. M. Lifshitz, |

6. | A. Sommerfeld, |

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

8. | A. Serdukov, I. Semchenko, S. Tretyakov, and A. Sihvola, |

9. | A. Lakhtakia, |

10. | A. Ishimaru, |

11. | J. A. Kong, |

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

13. | V. G. Veselago, “The electrodynamics of substances
with simultaneously negative values of |

14. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a
negative index of refraction,” Science |

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

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

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

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

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

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

21. | Q. Cheng and T. J. Cui, “Negative refractions and backward
waves in biaxially anisotropic chiral media,”
Opt. Express |

**OCIS Codes**

(000.2690) General : General physics

(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: December 20, 2006

Revised Manuscript: February 15, 2007

Manuscript Accepted: February 27, 2007

Published: April 12, 2007

**Citation**

Chao Zhang and Tie Jun Cui, "Spatial dispersion and energy in strong chiral medium," Opt. Express **15**, 5114-5119 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5114

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### References

- L. Pasteur, Researches on the molecular asymmetry of natural organic products, English translation of French original, (Alembic Club Reprints Vol. 14, pp. 1-46 1905), facsimile reproduction by SPIE in a 1990 book.
- A. J. Fresnel, in OEvres comple‘tes dAugustin Fresnel, H. d. Senarmont, E. Verdet, and L. Fresnel, eds., (Imprimerie imperiale, Paris, 1866), Vol. 1.
- W. A. Shurcliff and S. S. Ballard, Polarized light (Van Nostrand Co., Princeton, 1964).
- Eugene Hecht, Optics, 3rd Ed. (Addison-Wesley, 1998).
- L. D. Landau and E. M. Lifshitz, Electromagnetics of continous media, vol. 8 of Course of Theoretical Physics, 2nd edition, English, (Pergamon Press, 1984). [PubMed]
- A. Sommerfeld, Lectures on Theoretical Physics: Optics (Academic, New York, 1952).
- 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).
- A. Serdukov, I. Semchenko, S. Tretyakov, A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 2001).
- A. Lakhtakia, Beltrami Fields In Chiral media (World Scientific Publishing Co. Pte. Ltd., Singapore, 1994). [CrossRef]
- A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, Englewood Cliffs, NJ, 1991).
- J. A. Kong, Electromagnetic Wave Theory (Wiley, NY, 1986).
- 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]
- V. G. Veselago,"The electrodynamics of substances with simultaneously negative values of ε and µ," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- R. A. Shelby, D. R. Smith and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001). [CrossRef] [PubMed]
- Y. Jin and S. He, "Focusing by a slab of chiral medium," Opt. Express 13, 4974 (2005). [CrossRef] [PubMed]
- T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004). [CrossRef]
- 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]
- 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]
- Q. Cheng and T. J. Cui, "Negative refractions in uniaxially anisotropic chiral media," Phys. Rev. B 73, 113104 (2006). [CrossRef]
- Q. Cheng and T. J. Cui, "Negative refractions and backward waves in biaxially anisotropic chiral media," Opt. Express 14, 6322 (2006). [CrossRef] [PubMed]

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