## Handedness reversal of circular Bragg phenomenon due to negative real permittivity and permeability

Optics Express, Vol. 11, Issue 7, pp. 716-722 (2003)

http://dx.doi.org/10.1364/OE.11.000716

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### Abstract

When the real parts of the permittivity and the permeability dyadics of a structurally chiral, magnetic-dielectric material are reversed in sign, the circular Bragg phenomenon displayed by the material is proved here to suffer a change which indicates that the structural handedness has been, in effect, reversed. Additionally, reflection and transmission coefficients suffer phase reversal.

© 2002 Optical Society of America

## 1. Introduction

6. J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik **113**, 213–222 (2002). [CrossRef]

*circular Bragg phenomenon*, and underlies the many optical applications of these materials [4, 7

7. A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C **19**, 427–434 (2002). [CrossRef]

8. J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001). [CrossRef]

10. H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. **101**, 329–340 (1983). [CrossRef]

11. V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A **456**, 125–161 (2000). [CrossRef]

12. A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998). [CrossRef]

13. F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) **31**, 691–708 (1970). [CrossRef]

14. A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. **14**, 447–449 (2002). [CrossRef]

*left-handed materials*which are macroscopically homogeneous and display negative phase velocities, but are not chiral [16, 17].

*i*ω

*t*) time- dependence is implicit, with ω as the angular frequency; ∊

_{0}and µ

_{0}are the free-space permittivity and permeability, respectively;

*k*

_{0}=ω(∊

_{0}μ

_{0})

^{1/2}is the free-space wavenumber and λ

_{0}=2π/

*k*

_{0}is the free-space wavelength. A cartesian coordinate system is used, with

**u**

*,*

_{x}**u**

*and*

_{y}**u**

*as the cartesian unit vectors.*

_{z}## 2. Theory

*L*are given as

*T*denotes the transpose. The tilt dyadic

*S̳*=

_{y}**u**

_{y}**u**

*+(*

_{y}**u**

_{x}**u**

*+*

_{x}**u**

_{z}**u**

*)cosχ+(*

_{z}**u**

_{z}**u**

*-*

_{x}**u**

_{x}**u**

*) sinχ is a function of the angle χ ∈ [0, π/2]. The rotation dyadic*

_{z}*S̳*=

_{z}**u**

_{z}**u**

*+(*

_{z}**u**

_{x}**u**

*+u*

_{x}

_{y}**u**

*) cosζ+(*

_{y}**u**

_{y}**u**

*-*

_{x}**u**

_{x}**u**

*) sinζ, with ζ=*

_{y}*h*π

*z*/Ω, involves 2Ω as the structural period along the

*z*axis. The parameter

*h*is allowed only two values:+1 for structural right-handedness, and -1 for structural left-handedness. The complex-valued scalars ∊

*and µ*

_{a,b,c}*are functions of ω. For later use, ∊˜*

_{a,b,c}*=∊*

_{d}*a*µ

*a*/(∊

*a*cos

^{2}χ+∊

*sin*

_{b}^{2}χ) and µ˜

*=µ*

_{d}*µ*

_{a}*/(µ*

_{b}*cos*

_{a}^{2}χ+µ

*sin*

_{b}^{2}χ) are defined.

12. A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998). [CrossRef]

*f̱*(

*z*)]=[

*Ẽ*(

_{x}*z*),

*Ẽ*(

_{y}*z*),

*H̃*

_{x}(

*z*),

*H̃*

_{y}(

*z*)]

*is a column vector, while the 4×4 matrix function [*

^{T}*P̳*(

*z*)] is specified as follows:

11. V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A **456**, 125–161 (2000). [CrossRef]

12. A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998). [CrossRef]

18. M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) **188**, 1563–1575 (2001). [CrossRef]

*M̳*] is a 4×4 matrix.

*z*≤0 and

*z*≥

*L*are vacuous. Let the incident, reflected and transmitted plane waves be represented by

^{-1}(

*κ/K*

_{0}), while the unit vectors s=-

**u**

*sin ψ+*

_{x}**u**

*cos ψ and*

_{y}**p**±=∓(

**u**

*cosψ+*

_{x}**u**

*sin ψ) cosθ+*

_{y}**u**

*sin θ. The amplitudes of the LCP and the RCP components of the incident plane wave, denoted by*

_{z}*a*and

_{L}*a*, respectively, are assumed given. The four unknown amplitudes (

_{R}*r*,

_{L}*r*,

_{R}*t*and

_{L}*t*) of the circularly polarized components of the reflected and transmitted plane waves are determined by solving a boundary value problem obtained by enforcing the boundary conditions on the interfaces

_{R}*z*=0 and

*z*=

*L*and using (9)–(11) in (8). The result is best put in terms of reflection coefficients (

*r*, etc.) and transmission coefficients (

_{LL}*t*, etc.) appearing in the 2×2 matrixes on the right sides of the following relations:

_{LL}## 3. Discussion

*Negative Refraction and Metamaterials*of this special issue, the intention here is to present a very significant

*observable*consequence of the transformation

*R*=|

_{LL}*r*|

_{LL}^{2}, etc) shown in Fig. 1. The wavelength-range for this figure was chosen to focus on the circular-polarization-sensitivity of the circular Bragg phenomenon, by an examination of three cases:

*h*=+1) with all components of Re [∊͇(

**r**)] and Re [µ͇(

**r**)] positive, a high-reflectance ridge is evident in the plots of

*R*accompanied by virtually null-valued

_{RR}*R*and very small

_{LL}*R*=

_{RL}*R*.

_{LR}*h*=-1) with all components of Re [∊͇(

**r**)] and Re [µ͇(

**r**)] positive, the circular-polarization-sensitivity is reversed. A high-reflectance ridge is evident in the plot of

*R*accompanied by virtually null-valued

_{LL}*R*.

_{RR}**r**)] and Re [µ͇(

**r**)] are negative, the reflectance plots are exactly like that for case (ii).

*T*=|

_{LL}*t*|

_{LL}^{2}, etc.) corroborate these observations. Clearly, the transformation (13) amounts to a change in structural handedness. This effect was observed in all computational results at every λ

_{0}∈ [200, 2000] nm tried; and it had been reported earlier [14

14. A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. **14**, 447–449 (2002). [CrossRef]

*] and Re[µ*

_{a,b,c}*] is as follows: Close examination of the matrix [*

_{a,b,c}*P͇*(

*z*)] reveals the following symmetries:

*R͇*]=Diag[1, -1, -1, 1] and the asterisk denotes the complex conjugate. Substitution of the symmetries (14) and (15) in (3) leads to the following identities for the tangential components of the electromagnetic fields:

*effectively*reversed. Furthermore, a phase reversal of the reflection and transmission coefficients is also indicated.

*P͇*

_{1}(

*z*)] [0ʹ], 0≤

*z*≤

*L*, the reversal of the incident wave vector is not necessary for the handedness reversal of the circular Bragg phenomenon to be manifested exactly. This special condition holds for either normal incidence (i.e., θ=0) or cholesteric structure (i.e., χ=0), or both.

*κ*=0) and assuming the absence of dissipation. Then, the eigenmodes inside a chiral ferrosmectic slab are either left or right elliptically polarized, with their respective vibration ellipses rotating along the

*z*axis in accordance with the structural handedness of the material [21

21. S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. **97**, 42–50 (1995). [CrossRef]

*>0 and µ*

_{a,b}*>0, the direction of the phase velocity of a particular mode is the same as the (common) direction of energy transport. However, when ∊*

_{a,b}*<0 and µ*

_{a,b}*<0, not only does the phase velocity reverse in direction, but the handedness of the vibration ellipse also reverses, while the direction of energy flow as well as the sense of rotation of the vibration ellipse remain unchanged. The reversal of all four modal handednesses amounts to an effective reversal of the structural handedness. As [*

_{a,b}*P͇*(

*z*)] is a holomorphic function of

*k*, the foregoing understanding would hold even for oblique incidence conditions, by virtue of analytic continuation [22]. For the same reason, the understanding should hold for weak dissipation too.

*z*≤0 were occupied by an optically denser material than the chiral ferrosmectic slab, and if dissipation in that half-space were negligible, the phase reversal of the reflection coefficients would make the incident beam spring backwards on total reflection [23

23. A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics **23**, 71–75 (2003). [CrossRef]

## 4. Concluding remarks

*and µ*

_{a,b,c}*. Neither are the conclusions dependent of the variations of ∊*

_{a,b,c}*and µ*

_{a,b,c}*with the angular frequency ω. Therefore, the presented conclusions should hold without restrictions on anisotropy, dispersion and dissipation.*

_{a,b,c}## References and links

1. | N. Kato, “The significance of Ewald’s dynamical theory of diffraction,” in |

2. | H. A. Macleod, |

3. | I. J. Hodgkinson and Q. h. Wu, |

4. | S. D. Jacobs (ed), |

5. | V. C. Venugopal and A. Lakhtakia, “Sculptured thin films: Conception, optical properties, and applications,” in |

6. | J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik |

7. | A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C |

8. | J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001). [CrossRef] |

9. | http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/TimeBragg/TD Bragg.html |

10. | H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. |

11. | V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A |

12. | A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998). [CrossRef] |

13. | F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) |

14. | A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. |

15. | V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) |

16. | J. Pendry, “Electromagnetic materials enter the negative age,” Physics World14 (9), 47–51 (2001), September issue. |

17. | A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. |

18. | M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) |

19. | F. de Fornel, |

20. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

21. | S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. |

22. | P. M. Morse and H. Feshbach, |

23. | A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics |

**OCIS Codes**

(230.1480) Optical devices : Bragg reflectors

(260.1440) Physical optics : Birefringence

(260.2110) Physical optics : Electromagnetic optics

(310.6870) Thin films : Thin films, other properties

**ToC Category:**

Focus Issue: Negative refraction and metamaterials

**History**

Original Manuscript: January 17, 2003

Revised Manuscript: February 24, 2003

Published: April 7, 2003

**Citation**

Akhlesh Lakhtakia, "Handedness reversal of circular Bragg phenomenon due to negative real permittivity and permeability," Opt. Express **11**, 716-722 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-716

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### References

- N. Kato, �??The signi.cance of Ewald�??s dynamical theory of diffraction,�?? in P.P. Ewald and His Dynamical Theory of X-ray Diffraction (D. W. J. Cruickshank, H. J. Juretschke and N. Kato, eds) (Oxford University Press, Oxford, UK, 1992), pp. 3-23.
- H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185-208.
- I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302-322.
- S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIEO ptical Engineering Press, Bellingham, WA, USA, 1992).
- V. C. Venugopal and A. Lakhtakia, �??Sculptured thin films: Conception, optical properties, and applications,�?? in Electromagnetic Fields in Unconventional Materials and Structures (O. N. Singh and A. Lakhtakia, eds) (Wiley, New York, NY, USA, 2000), pp. 151-216.
- J. Wang, A. Lakhtakia and J. B. Geddes III, �??Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,�?? Optik 113, 213-222 (2002). [CrossRef]
- A. Lakhtakia, �??Sculptured thin films: accomplishments and emerging uses,�?? Mater. Sci. Eng. C 19, 427-434 (2002). [CrossRef]
- J. B. Geddes III and A. Lakhtakia, �??Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,�?? Eur. Phys. J. Appl. Phys. 13, 3-14 (2001); corrections: 16, 247 (2001). [CrossRef]
- <a href="http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/TimeBragg/TD Bragg.html">http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/TimeBragg/TD Bragg.html</a>
- H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda and E. Kuze, �??Experimental study on higher order re.ection by monodomain cholesteric liquid crystals,�?? Mol. Cryst. Liq. Cryst. 101, 329-340 (1983) [CrossRef]
- V. C. Venugopal and A. Lakhtakia, �??Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,�?? Proc. R. Soc. Lond. A 456, 125-161 (2000). [CrossRef]
- A. Lakhtakia and W. S. Weiglhofer, �??Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,�?? Proc. R. Soc. Lond. A 453, 93-105 (1997); corrections: 454, 3275 (1998). [CrossRef]
- F. Brochard and P.G. de Gennes, �??Theory of magnetic suspensions in liquid crystals,�?? J. Phys. (Paris) 31, 691-708 (1970). [CrossRef]
- A. Lakhtakia, �??Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,�?? Adv. Mater. 14, 447-449 (2002). [CrossRef]
- V. Ponsinet, P. Fabre, M. Veyssie and L. Auvray, �??A small-angle neutron-scattering study of the ferrosmectic phase,�?? J. Phys. II (Paris) 3, 1021-1039 (1993).
- J. Pendry, �??Electromagnetic materials enter the negative age,�?? Phys. World 14 (9), 47-51 (2001), September issue.
- A. Lakhtakia, M. W. McCall and W. S. Weiglhofer, �??Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),�?? Arch. Elektr. Uber. 56, 407- 410 (2002).
- M. Schubert and C. M. Herzinger, �??Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,�?? Phys. Stat. Sol. (a) 188, 1563-1575 (2001). [CrossRef]
- F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12-18.
- H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell Syst. Tech. J. 48, 2909-2947 (1969).
- S. F. Nagle, A. Lakhtakia and W. Thompson, Jr., �??Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),�?? J. Acoust. Soc. Am. 97, 42-50 (1995). [CrossRef]
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3.
- A. Lakhtakia, �??On planewave remittances and Goos-Hanchen shifts of planar slabs with negative real permittivity and permeability,�?? Electromagnetics 23, 71-75 (2003). [CrossRef]

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