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

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. c © 2003 Optical Society of America OCIS codes: (230.1480) Bragg reflectors; (260.1440) Birefringence; (260.2110) Electromagnetic theory; (310.6870) Thin films, other properties References and links 1. N. Kato, “The significance 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. 2. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185-208. 3. I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302-322. 4. S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIE Optical Engineering Press, Bellingham, WA, USA, 1992). 5. 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. 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). 7. A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C 19, 427-434 (2002). 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). 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. 101, 329-340 (1983). 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). 12. 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). 13. F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691-708 (1970). 14. A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. 14, 447-449 (2002). 15. 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). 16. J. Pendry, “Electromagnetic materials enter the negative age,” Physics World 14 (9), 47-51 (2001), September issue. (C) 2003 OSA 7 April 2003 / Vol. 11, No. 7 / OPTICS EXPRESS 716 #1994 $15.00 US Received January 17, 2003; Revised February 24, 2003 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. 56, 407410 (2002). 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). 19. F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12-18. 20. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969). 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). 22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3. 23. A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23, 71-75 (2003).


Introduction
Constructive and destructive interference of the consequences of electromagnetic processes occuring in different parts of a periodically nonhomogeneous material is responsible for the Bragg phenomenon.From a mathematical viewpoint, the interaction of the spatial harmonics of the electromagnetic field and the spatial harmonics of the permiittivity (or any other relevant constitutive property) can become resonant [1], thereby leading to virtually total reflection in certain, fairly narrow, wavelength-regimes.
The Bragg phenomenon is widely exploited for filtering by cascades of alternating homogeneous layers of two materials of different permittivities [2].Provided that both types of layers are isotropic, and light is normally incident, the Bragg phenomenon is insensitive to the polarization state of the incident light.That insensitivity generally vanishes with the introduction of anisotropy (although special cases of virtually no sensitivity can still be conceived of [3]).
A particularly interesting feature -discrimination between incident left and right circularly polarized (LCP and RCP) plane waves -emerges when the Bragg phenomenon is displayed by chiral liquid crystals [4] and chiral sculptured thin films [5,6].These anisotropic materials are structurally chiral (i.e., handed) and unidirectionally nonhomogeneous.When the wave vector of the incident plane wave is parallel to the direction of nonhomogeneity, extremely high reflection is observed if the handedness of the incident light coincides with the structural handedness of the material; and extremely low reflection is observed if the two handednesses do not match.This circularpolarization-sensitivity gives rise to the term circular Bragg phenomenon, and underlies the many optical applications of these materials [4,7].In the time domain, the circular-polarization-sensitive high reflection manifests itself as a pulse-bleeding phenomenon [8] -which becomes clear from viewing movies of a pulse moving across the planar interface of a structurally chiral material and free space [9].The circular Bragg phenomenon (accompanied by less significant analogs in one lower and many higher frequency regimes) also occurs for incidence in other directions [10,11].
Structurally chiral materials can have magnetic properties in addition to dielectric [12].For normal incidence on slabs made of the simplest of these type of materials with both magnetic and dielectric properties -called ferrocholesteric materials [13] -the circular Bragg phenomenon was recently shown [14] to reverse its circular-polarizationsensitivity when the real parts of its permeability and permittivity dyadics are changed from positive to negative.That observation is extended here for oblique incidence on chiral ferrosmectic slabs [15], a far more general scenario; and a mathematical proof is provided.This work is obviously inspired by the recent spate of papers published on the inappropriately designated left-handed materials which are macroscopically homo-geneous and display negative phase velocities, but are not chiral [16,17].
A note about notation: Vectors are in boldface, dyadics are double underlined, column vectors are underlined and enclosed in square brackets, while matrixes are double underlined and also enclosed in square brackets.An exp(−iωt) 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 y and u z as the cartesian unit vectors.

Theory
The nonhomogeneous permittivity and the permeability dyadics of a chiral ferrosmectic slab of thickness L are given as (1) where the superscript T denotes the transpose.The tilt dyadic S y = u y u y + (u 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 a,b,c and µ a,b,c are functions of ω.For later use, Without loss of generality, the field phasors everywhere can be written as where κ ∈ [0, ∞) and ψ ∈ [0 , 2π].The fields inside the chiral ferrosmectic slab must follow the 4×4 matrix ordinary differential equation [12] In this equation, f (z) = Ẽx (z), Ẽy (z), Hx (z), Hy (z) T is a column vector, while the 4×4 matrix function P (z) is specified as follows: Equation ( 3) can be solved in a variety of ways, mostly numerical [11,12,18].The result can be stated in matrix form as where M is a 4×4 matrix.Suppose the half-spaces z ≤ 0 and z ≥ L are vacuous.Let the incident, reflected and transmitted plane waves be represented by respectively, where θ = sin −1 (κ/k 0 ), while the unit vectors s = −u x sin ψ + u y cos ψ and p ± = ∓ (u x cos ψ + u y sin ψ) cos θ + u z sin θ.The amplitudes of the LCP and the RCP components of the incident plane wave, denoted by a L and a R , respectively, are assumed given.The four unknown amplitudes (r L , r R , t L and t R ) 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 z = 0 and z = L and using (9)-( 11) in (8).The result is best put in terms of reflection coefficients (r LL , etc.) and transmission coefficients (t LL , etc.) appearing in the 2×2 matrixes on the right sides of the following relations:

Discussion
Consistently with the theme Negative Refraction and Metamaterials of this special issue, the intention here is to present a very significant observable consequence of the transformation This consequence is evident in the plots of the reflectances (R LL = |r LL | 2 , etc) shown in Fig. 1.The wavelength-range for this figure was chosen to focus on the circularpolarization-sensitivity of the circular Bragg phenomenon, by an examination of three cases: (i) For a structurally right-handed ferrosmectic slab (h = +1) with all components of Re (r) and Re µ(r) positive, a high-reflectance ridge is evident in the plots of R RR accompanied by virtually null-valued R LL and very small R RL = R LR .
(ii) For a structurally left-handed ferrosmectic slab (h = −1) with all components of Re (r) and Re µ(r) positive, the circular-polarization-sensitivity is reversed.A high-reflectance ridge is evident in the plot of R LL accompanied by virtually null-valued R RR .
(iii) Finally, when the chosen material is structurally right-handed but all components of Re (r) and Re µ(r) are negative, the reflectance plots are exactly like that for case (ii).
The transmittances (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] for the special case θ = χ = 0.A general proof for the handedness reversal of the circular Bragg phenomenon by virtue of a reversal of the signs of Re [ a,b,c ] and Re [µ a,b,c ] is as follows: Close examination of the matrix P (z) reveals the following symmetries: where the 4×4 diagonal matrix R = 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: Application of the identity ( 16) to ( 9)-( 11) suggests the relationship while the identity (17) implies that The following two relationships thereby emerge: The foregoing analysis thus proves that if (i) the signs of the real parts of the permittivity and permeability dyadics of a chiral ferrosmectic slab are reversed, and (ii) the transverse components of incident wave vector are also reversed, then the reflectances and the transmittances indicate that the structural handedness has been effectively reversed.Furthermore, a phase reversal of the reflection and transmission coefficients is also indicated.
With the relationships ( 20) and ( 21) involving all components of the permittivity and the permeability dyadics, no definite statement on handedness reversal can be extracted therefrom if not all components of the two dyadics suffer a change of sign in their real parts.
The reflection and transmission coefficients being weakly dependent on ψ [5], calculations show that the handedness reversal of the circular Bragg phenomenon is evident, although approximately, even if the transverse components of the incident wave vector are not reversed.However, when P 1 (z) ≡ 0 , 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.
As a chiral ferrosmectic medium is continuously nonhomogeneous, general physical pictures of the circular Bragg phenomenon are difficult to develop, in contrast to their piecewise homogeneous counterparts [20].Some understanding does become possible on analysis for normal incidence (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].When a,b > 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 P (z) is a holomorphic function of κ, 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.
A consequence of the phase reversal of the reflection coefficients due to the transformation ( 13) are negative Goos-Hänchen shifts [19].When conditions for total reflection prevail at a planar interface, a beam of finite width is seen to spring forward a certain distance on reflection.This distance is called the Goos-Hänchen shift.If the half-space 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].Causality, however, would not be violated.

Concluding remarks
When the real parts of the permittivity and the permeability dyadics of a chiral ferrosmectic slab are reversed in sign, the circular Bragg phenomenon displayed is proved here to suffer a change that indicates the effective reversal of structural handedness.Furthermore, reflection and transmission coefficients suffer phase reversal, which indicates the exhibition of negative Goos-Hänchen shifts when appropriate conditions prevail.
These conclusions are independent of the magnitudes of the real and imaginary parts of a,b,c 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.