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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 7 — Apr. 7, 2003
  • pp: 716–722
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Handedness reversal of circular Bragg phenomenon due to negative real permittivity and permeability

Akhlesh Lakhtakia  »View Author Affiliations


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

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

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.

], 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

2. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185–208.

]. 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

3. I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302–322.

]).

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

4. S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIEO ptical Engineering Press, Bellingham, WA, USA, 1992).

] and chiral sculptured thin films [5

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

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]

]. 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 circular-polarization-sensitivity gives rise to the term circular Bragg phenomenon, and underlies the many optical applications of these materials [4

4. S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIEO ptical Engineering Press, Bellingham, WA, USA, 1992).

, 7

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

]. In the time domain, the circular-polarization-sensitive high reflection manifests itself as a pulse-bleeding phenomenon [8

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]

] — 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

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

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]

].

Structurally chiral materials can have magnetic properties in addition to dielectric [12

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]

]. For normal incidence on slabs made of the simplest of these type of materials with both magnetic and dielectric properties — called ferrocholesteric materials [13

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

] — the circular Bragg phenomenon was recently shown [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]

] to reverse its circular-polarization-sensitivity 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

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).

], 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 homogeneous and display negative phase velocities, but are not chiral [16

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

, 17

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, 407–410 (2002).

].

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 µ0 are the free-space permittivity and permeability, respectively; k 0=ω(∊0μ0)1/2 is the free-space wavenumber and λ0=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.

2. Theory

The nonhomogeneous permittivity and the permeability dyadics of a chiral ferrosmectic slab of thickness L are given as

¯¯(r)=0S¯¯zS¯¯y[auzuz+buxux+cuyuy]S¯¯yTS¯¯zTμ¯¯(r)=μ0S¯¯zS¯¯y[μauzuz+μbuxux+μcuyuy]S¯¯yTS¯¯zT},0zL,
(1)

where the superscript T denotes the transpose. The tilt dyadic y=u y u y+(u x u x+u z u z)cosχ+(u z u x-u x u z) sinχ is a function of the angle χ ∈ [0, π/2]. The rotation dyadic z=u z u z+(u x u x+uy u y) cosζ+(u y u x-u x u y) sinζ, with ζ=hπz/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 ∊a,b,c and µa,b,c are functions of ω. For later use, ∊˜d=∊aµa/(∊a cos2 χ+∊b sin2 χ) and µ˜daµb/(µa cos2 χ+µb sin2 χ) are defined.

Without loss of generality, the field phasors everywhere can be written as

E(r)=E˜(z)exp[(xcosψ+ysinψ)]H(r)=H˜(z)exp[(xcosψ+ysinψ)]},<z<,
(2)

where κ ∈ [0, ∞) and ψ ∈ [0, 2π]. The fields inside the chiral ferrosmectic slab must follow the 4×4 matrix ordinary differential equation [12

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]

]

ddz[f¯(z)]=i[P¯¯(z)][f¯(z)],0<z<L.
(3)

In this equation, [(z)]=[x(z), y(z), x(z), y(z)]T is a column vector, while the 4×4 matrix function [(z)] is specified as follows:

[P¯¯(z)]=[P¯¯0(z)]+κk0[P¯¯1(z)]+(κk0)2[P¯¯2(z)],
(4)
[P¯¯0(z)]=ω{[000μ0μc+μ˜d200μ0μc+μ˜d2000c+˜d2000c+˜d2000]
+[00μ0μcμ˜d2sin2ζμ0μcμ˜d2cos2ζ00μ0μcμ˜d2cos2ζμ0μcμ˜d2sin2ζ0c˜d2sin2ζ0c˜d2cos2ζ000c˜d2cos2ζ0c˜d2sin2ζ00]},
(5)
[P¯¯1(z)]=(k0sinχcosχ)×
{˜d(ab)ab[cosζcosψ0000sinζsinψ0000sinζsinψ0000cosζcosψ]
+μ˜d(μaμb)μaμb[sinζsinψ0000cosζcosψ0000cosζcosψ0000sinζsinψ]
+˜dμ˜d(aμbbμa)abμaμb[0sinζcosψ00cosζsinψ000000sinζcosψ00cosζsinψ0]},
(6)
[P¯¯2(z)]=ω×
[00μ0˜dabcosψsinψμ0˜dabcos2ψ00μ0˜dabsin2ψμ0˜dabcosψsinψ0μ˜dμaμbcosψsinψ0μ˜dμaμbcos2ψ000μ˜dμaμbsin2ψ0μ˜dμaμbcosψsinψ00].
(7)

Equation (3) can be solved in a variety of ways, mostly numerical [11

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

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

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]

]. The result can be stated in matrix form as

[f¯(L)]=[M¯¯][f¯(0)],
(8)

where [] is a 4×4 matrix.

Suppose the half-spaces z≤0 and zL are vacuous. Let the incident, reflected and transmitted plane waves be represented by

einc(r)=[(isp+)2aL(is+p+)2aR]eik0zcosθ,z0,
(9)
eref(r)=[(isp)2rL+(is+p)2rR]eik0zcosθ,z0,
(10)
etr(r)=[(isp+)2tL(is+p+)2tR]eik0(zL)cosθ,zL,
(11)

respectively, where θ=sin-1(κ/K 0), while the unit vectors s=-u x sin ψ+u ycos ψ and p±=∓(u x cosψ+u y sin ψ) cosθ+u z sin θ. The amplitudes of the LCP and the RCP components of the incident plane wave, denoted by aL and aR, respectively, are assumed given. The four unknown amplitudes (rL, rR, tL and tR) 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 (rLL, etc.) and transmission coefficients (tLL, etc.) appearing in the 2×2 matrixes on the right sides of the following relations:

[rLrR]=[rLLrLRrRLrRR][aLaR],[tLtR]=[tLLtLRtRLtRR][aLaR].
(12)
Fig. 1. Computed reflectances of a chiral ferrosmectic slab in free space, for Ω=140 nm, L=60Ω, and χ=30°. Case (i): h=1, ψ=35°, ∊a=2.7(1+iδ), ∊b=3.3(1+iδ), ∊c=3(1+iδ), µa=1.1(1+iδµ), µb=1.4(1+iδµ), µc=1.2(1+iδµ), δ=2δµ=2× 10-3. Case (ii): Same as (i) except h=-1 and ψ=-35°. Case (iii): Same as (i) except that ψ=215° and the real parts of ∊a,b,c and μa,b,c are negative.

3. 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

{Re[¯¯(r)]Re[¯¯(r)],Re[μ¯¯(r)]Re[μ¯¯(r)],0zL}.
(13)

This consequence is evident in the plots of the reflectances (RLL=|rLL|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:

(i) For a structurally right-handed ferrosmectic slab (h=+1) with all components of Re [∊͇(r)] and Re [µ͇(r)] positive, a high-reflectance ridge is evident in the plots of RRR accompanied by virtually null-valued RLL and very small RRL=RLR.

(ii) For a structurally left-handed ferrosmectic slab (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 RLL accompanied by virtually null-valued RRR.

(iii) Finally, when the chosen material is structurally right- handed but all components of Re [∊͇(r)] and Re [µ͇(r)] are negative, the reflectance plots are exactly like that for case (ii).

The transmittances (TLL=|tLL|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 λ0 ∈ [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]

] 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[µa,b,c] is as follows: Close examination of the matrix [(z)] reveals the following symmetries:

z[0,L],[P¯¯(z;¯¯(r),μ¯¯(r);h,ψ)]=[R¯¯][P¯¯(z;¯¯(r),μ¯¯(r);h,ψ)][R¯¯]
(14)
=[P¯¯(z;¯¯*(r),μ¯¯*(r);h,π+ψ)]*,
(15)

where the 4×4 diagonal matrix []=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:

z0,L,[f¯(z;¯¯(r),μ¯¯(r);h,ψ)]=[R¯¯][f¯(z;¯¯(r),μ¯¯(r);h,ψ)]
(16)
=[f¯(z;¯¯*(r),μ¯¯*(r);h,π+ψ)]*.
(17)

Application of the identity (16) to (9)-(11) suggests the relationship

{hh,ψψ}{aLaR,rLrR,tLtR};
(18)

while the identity (17) implies that

{Re[a,b,c]Re[a,b,c]Re[μa,b,c]Re[μa,b,c]ψπ+ψ}{aLaR*,aRaL*rLrR*,rRrL*tLtR*,tRtL*}.
(19)

The following two relationships thereby emerge:

{hh,ψψ}{rLLrRR,rLRrRLtLLtRR,tLRtRL},
(20)
{Re[a,b,c]Re[a,b,c]Re[μa,b,c]Re[μa,b,c]ψπ+ψ}
{rLLrRR*,rRRrLL*,rLRrRL*,rRLrLR*tLLtRR*,tRRtLL*,tLRtRL*,tRLtLR*}.
(21)

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

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.

], 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 [ 1(z)] [0ʹ], 0≤zL, 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.

A consequence of the phase reversal of the reflection coefficients due to the transformation (13) are negative Goos-Hänchen shifts [19

19. F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12–18.

]. 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

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]

]. Causality, however, would not be violated.

4. 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 µa,b,c. Neither are the conclusions dependent of the variations of ∊a,b,c and µa,b,c with the angular frequency ω. Therefore, the presented conclusions should hold without restrictions on anisotropy, dispersion and dissipation.

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 (SPIEO ptical 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). [CrossRef]

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]

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). [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]

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 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. 56, 407–410 (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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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

  1. 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.
  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 (SPIEO ptical 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). [CrossRef]
  7. A. Lakhtakia, �??Sculptured thin films: accomplishments and emerging uses,�?? Mater. Sci. Eng. 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]
  9. <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>
  10. 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]
  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. A 453, 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]
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