Defect modes in multisection helical photonic crystals

We examine the two defect modes in four–section helical photonic crystals (HPCs) that comprise three twist defects located at the intersections. The three twist defects are introduced by a single angle φ t , but they are quantified differently by the jump angles across the successive sections. The two defect modes are localized at the different defect sites and can be either coupled or uncoupled to each other, depending on the value of φt . Both defect modes are excited by normally incident plane waves of different circular polarization states as the HPC thickness increases. When the two defect modes are uncoupled to each other, two co–handed reflection holes are present in the Bragg regime for small thickness, but they evolve into two stable cross–handed transmission holes for sufficiently large thickness. When the two defect modes are coupled to each other, however, three co–handed reflection holes appear around the center of the Bragg regime for small thickness, and they evolve into three cross–handed transmission holes as the thickness increases, and eventually all three co–handed transmission holes coalesce into one stable cross–handed transmission hole for sufficiently large thickness. Finally, the simultaneous occurrence of the two types of spectral holes at a single resonance wavelength can be realized for specific values of sample thickness when the two defect modes are uncoupled to each other. © 2005 Optical Society of America OCIS codes: (160.1190) Anisotropic optical material; (230.3720) Liquid–crystal devices; (230.5440) Polarization–sensitive devices, (360.6860) Thin films, optical properties. References and links 1. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, (Clarendon Press, Oxford, UK, 1993). 2. Selected Papers on Liquid Crystals for Optics, S. D. 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Sherwin, Q. H. Wu, and I. J. Hodgkinson, “Sculptured–thin–film spectral holes for optical sensing of fluids,” Opt. Commun. 194, 33–46 (2001). 21. M. W. McCall and A. Lakhtakia, “Explicit expressions for spectral remittances of axially excited chiral sculptured thin films,” J. Mod. Opt. 51, 111–127 (2004). 22. F. Wang and A. Lakhtakia, “Optical crossover phenomenon due to a central 90◦–twist defect in a chiral sculptured thin film or chiral liquid crystal,” Proc. R. Soc. Lond. A 461, 2985–3004 (2005). 23. A. Lakhtakia, “Shear axial modes in a PCTSCM. Part VI: simpler transmission spectral holes,” Sens. Actuators A 87, 78–80 (2000). 24. I. J. Hodgkinson, A. Lakhtakia, Q.h. Wu, L. De Silva, and M.W. McCall, “Ambichiral, equichiral and finely chiral layered structures,” Opt. Commun. 239, 353–358 (2004).


Introduction
Periodic structurally chiral materials have been extensively studied during the last century due to the unique properties and applications of self-organized cholesteric liquid crystals (CLCs) [1,2].Recently, chiral sculptured thin films (STFs) have been grown by directional physical vapor deposition of a wide range of source materials as solid analogs of liquid crystals [3]- [5].Both CLCs and chiral STFs can be considered as helical photonic crystals, a term initiated by Becchi et al. [6] Most significantly, helical photonic crystals (HPCs) display the circular Bragg phenomenon and therefore are capable of discriminating between incident plane waves of different circular polarization states as follows [7]: Both HPCs and circularly polarized (CP) plane waves are handed, the former structurally and the latter in terms of the sense of rotation of the vibration ellipse.The hallmark of the circular Bragg phenomenon is that the reflectance of a co-handed normally incident CP plane wave -i.e., the structural handedness of the HPC is the same as the handedness of the vibration ellipse of the incident plane wave -is very high in the Bragg regime, provided the HPC is sufficiently thick and the wavelength lies in the so-called Bragg regime.If the incident plane wave's handedness is opposed to the structural handedness, i.e., for cross-handed incidence, the reflectance in the Bragg regime is very low.
Exploitation of the circular Bragg phenomenon for the design of CLC-and STF-based, CP-sensitive optical devices -specifically, polarization filters -has been successful in the past and remains an active area of research [2,5].
In contrast to the optical responses of regular, defect-free HPCs, a CP localized defect mode has been recently identified for HPCs with central phase defects [8]- [13].Defects can be exploited for making narrowband filters [14].A phase defect can be of two types: either a homogeneous layer [8,9] or a twist [11].Combinations of the layer and the twist defects may be advantageous in certain circumstances [13].Our focus here is on twist defects.
The defect mode is localized to the defect site and can be excited by a normally incident CP plane wave of either the co-handed type or the cross-handed type -depending on the thickness of the HPC.The defect mode displays the following crossover phenomenon [11]- [15]: When the HPC thickness is relatively small, the co-handed reflection spectrum is pierced by a hole occurring in the Bragg regime.As the thickness increases, the co-handed reflection hole diminishes and eventually vanishes, while a hole emerges in the cross-handed transmission spectrum.
The occurrence of the cross-handed transmission hole implies that the localized defect mode generates a photonic band gap in the Bragg regime [11,16].The width of the photonic band gap is tiny but stable for large thickness of the HPC, because the electromagnetic energy density at the defect site saturates instead of scaling exponentially with respect to the thickness [11].This defect mode can be harnessed for new optical devices, such as low threshold lasers [17], chiral filters [10], and chiral fibers [18], provided that dissipation in the HPC is small enough to be ignored [12].
When multisection HPCs are formed by cascading [19], multiple phase defects of different types can be introduced [20].Therefore, different CP defect modes can be localized at different resonance wavelengths.It is reasonable to conjecture that the thickness-dependent variability of an isolated defect mode manifesting either as a co-handed reflection hole or as a cross-handed transmission hole should endow couplings between different defect modes with different types of manifestation.
In this paper, we present the characteristics of two CP defect modes in a four-section HPC which has three intersectional twist defects, all described by an angle φ t .All four sections are of identical thickness.The two defect modes are localized at different defect sites and can be either uncoupled or coupled to each other, depending on the value of φ t .When uncoupled, the two defect modes manifest as two distinct cross-handed transmission holes for large HPC thickness; when coupled, only one such hole is produced.Additionally, a remarkable peculiarity of the two defect modes, when uncoupled, is that one of them can be excited by both the co-handed and the cross-handed incident CP plane waves, for specific values of section thickness.Therefore, the simultaneous occurrence of the two types of spectral holes in the spectrum of the same device can be possibly engineered, thereby enhancing the efficiency of optical transmission systems.
The organization of this paper is as follows: Section 2 describes the boundary value problem relevant to a CP plane wave normally incident on a four-section HPC with three intersectional twist defects.A coupled-wave theory is described in Section 3 to identify different regimes of φ t for defect-mode classification.The two CP defect modes -as manifested by the localized distribution of energy density inside the four-section HPC -are identified and characterized for two different regimes of φ t in Section 4. Our conclusions are summarized in Section 5, wherein is also discussed the extension to a 2 -section ( ≥ 2) HPC with 2 − 1 intersectional twist defects.
Fig. 1.Schematic of the boundary value problem involving a four-section HPC with three intersectional twist defects.The half-spaces z ≤ 0 and z ≥ 4D are filled with a homogeneous, isotropic, dielectric medium of refractive index n hs .The HPC is excited by a normally incident CP plane wave from the half-space z ≤ 0.

Boundary Value Problem
Suppose the region 0 < z < 4D is occupied by a four-section HPC, where D is the thickness of each section; while the half-spaces z ≤ 0 and z ≥ 4D are filled with a homogenous, isotropic, dielectric medium of refractive index n hs , as shown in Fig. 1.The relative permittivity dyadic of the four-section HPC is stated as follows: Here and hereafter, ε a,b,c are the reference relative permittivity scalars, and (u x , u y , u z ) are the Cartesian unit vectors with u z parallel to the axis of nonhomogeneity of the four-section HPC.
The tilt dyadic represents the locally aciculate morphology of each section of the four-section HPC, with χ ∈ [0, π/2] as the tilt angle.For a CLC, we have to set χ = 0 and ε a = ε c = ε b ; whereas χ > 0 and ε a = ε b = ε c in general for chiral STFs.
The rotation dyadic indicates the helical morphology of HPCs.Here, 2Ω is the structural period; ϕ j is the structural rotation angle of the j-th section, (1 ≤ j ≤ 4); and the parameter h = 1 for structural righthandedness and h = −1 for structural left-handedness.When ϕ j = ϕ j+1 , an intersectional twist defect is located at z = jD, (1 ≤ j ≤ 4), the twist defect being quantified by the jump angle ϕ j+1 − ϕ j . ( then, the three twist defects are as follows: All three twist defects can thus be ascribed to the single angle φ t , but they can be classified as either the φ t -twist defect or the −φ t -twist defect, in accord with (5), for the purpose of discussion.We also note that an isolated −φ t -twist defect is the same as an isolated (π − φ t )twist defect, in the present context.Generally, the three scalars ε a,b,c are implicitly dependent on the free-space wavelength λ 0 , but here we ignore those dependences because our attention is confined to narrow wavelengthregimes.Furthermore, we also ignore any dissipation, so that ε a,b,c are all real-valued and positive.For convenience, we define the composite relative permittivity scalar and the normalized thickness The four-section HPC is axially excited by a plane wave that is normally incident from the half-space z ≤ 0. Therefore, a plane wave is reflected into the half-space z ≤ 0 and another is transmitted into the half-space z ≥ 4D.The electric field phasors associated with the incident, reflected, and transmitted plane waves are stated as and respectively, where u ± = (u x ± iu y )/ √ 2; k 0 = 2π/λ 0 is the free-space wavenumber; a L and a R are the known amplitudes of the left and right CP components of the incident plane wave; r L and r R are the unknown amplitudes of the reflected planewave components; and t L and t R are the unknown amplitudes of the transmitted planewave components.
The procedure to obtain the planewave reflectances and transmittances is devised from the solution of a boundary value problem detailed elsewhere [5,8].Let us content ourselves here by stating that 4 × 4 algebraic matrix equation [8] [ eventually emerges, where the column-4 vectors [ f entry ] and [ f exit ] contain the x-and the ycomponents of the electric and magnetic field phasors at the z = 0 and z = 4D planes, respectively.In accordance with ( 8)- (10), where η 0 is the intrinsic impedance of free space.The 4 × 4 matrix [M] is a product of four sectional matrixes as where and The planewave remittances (i.e., reflectances and transmittances) and the electromagnetic field phasors E(z) and H(z) in the region 0 < z < 4D can be computed after numerically solving (11) to obtain {r L , r R } and {t L ,t R } in terms of {a L , a R }.Thereafter, the four reflectances R LL , etc., and the four transmittances T LL , etc., can be determined as follows: Remittances with two different subscripts are called cross-polarized, and those with both subscripts identical are called co-polarized.Cross-polarized remittances can be minimized by index-matching across the planes z = 0 and z = 4D [5,13]; and in this paper we therefore set Co-polarized remittances subscripted RR (LL) are dubbed as co-handed and those subscripted LL (RR) are dubbed cross-handed, when the four-section HCP is right(left)-handed.The timeaveraged energy density can also be computed in the region 0 < z < 4D, with ε 0 and μ 0 being the permittivity and the permeability of free space, the asterisk denoting the complex conjugate, and Re meaning the

Coupled-wave Theory
Before proceeding to a discussion of numerical results, it is advantageous to use the approximate but analytically tractable coupled-wave theory (CWT) for axially excited HPCs.A complete statement of the CWT for axial wave propagation in defect-free HPCs is detailed elsewhere [21] and has been generalized to analyze the crossover phenomenon in two-section HPCs [22].Let us begin by considering a single-section HPC.Ignoring the index-mismatch at its entry and exit pupils, we can relate the CP components of the electric field phasors at the entry and exit pupils as 1[ where and the 4 × 4 transfer matrix Here, E + L and E + R , respectively, denote the left and right CP components of the electric field phasor of the forward propagating wave, while E − L and E − R , respectively, denote the left and right CP components of the electric field phasor of the backward propagating wave.The quantities and where The transfer matrix of a multisection HPC must be a sequential product of the individual transfer matrixes of each sections.For the four-section HPC described in Section 2, the transfer matrix where ] by a rotational transformation as follows: The 4 × 4 rotational matrix consists of two diagonal blocks which employ the 2×2 matrixes and 0 is the 2 × 2 null matrix; and the superscript † denotes the Hermitian adjoint.When ν D is small (but still large enough for the circular Bragg phenomenon to be fully manifested), co-handed reflection holes emerge in the Bragg regime for the four-section HPC, which corresponds to the fact that the transfer matrix [W hpc ] turns out to be equal to a 4 × 4 diagonal matrix at certain wavelengths. 2In other words, at these resonance wavelengths, the four-section HPC acts as a homogeneous, isotropic, dielectric medium of refractive index n for normally incident CP plane waves, whether co-handed or cross-handed [22].The requirement of [W hpc ] being diagonal leads to the equalities P+ + e i2φ t P * + = 0 (33) where and δ u,v is the Kronecker delta.
For convenience, let us define Accordingly, a wavelength-regime can be mapped to a β -regime uniquely; for example, β = 0 corresponds to the center-wavelength of the Bragg regime of a defect-free HPC [5].Three values of β are important.First, (33) yields the value next, (34) yields the values Normally, the three values of β -namely, β 0 and β ± -correspond to three different resonance wavelengths.However, β − → β 0 as |cosφ t |/δ φ t → ∞, where Then, only two resonance wavelengths can be identified, one for β + and the other for β − = β 0 .Because δ φ t << 1 for full development of the circular Bragg phenomenon [5], (40) leads to Table 1 shows the values of δ φ t for various values of ν D .Clearly, δ φ t → 0 rapidly as ν D increases.
• an intermediate regime.
Since ν D must be large for the full development of the circular Bragg phenomenon, from Table 1 it is clear that δ φ t ≈ 0 for cases of practical interest; the intermediate regime is then of little significance, and is therefore ignored in the remainder of this paper.

Numerical Results and Discussion
The angle φ t ∈ (0, π) suffices to describe the chosen four-section HPC.The number of spectral holes in the remittance spectrums depends on the value of |π/2 − φ t |, as discussed next in Sections 4.1 and 4.2.We recall here that δ φ t is a very small angle because ν D is sufficiently large.
For this case, the optical response of the four-section HPC is exemplified by Fig. 2 for φ t = π/3.Clearly, two spectral holes are present in this figure, which are located at two distinct resonance wavelengths λ 0 1 < λ 0 2 , and their characteristics evolve with increasing ν D .In general, two cohanded reflection holes occur for small ν D , but both wane and eventually vanish, being replaced by two cross-handed transmission holes for sufficiently large ν D .Fig. 2 and the expressions (38) and (39) for β 0 and β ± permit us to deduce and  2. Spectrums of the co-handed transmittance T RR (solid lines) and the cross-handed transmittance T LL (dashed lines), computed for a four-section HPC with ε a = 2.62, ε b = 3.18, ε c = 2.72, χ = π/6, (thereby ε d = 3.02), h = 1, φ t = π/3, and Ω = 200 nm.The refractive index n hs = (ε c + ε d )/2 was chosen to diminish the index-mismatch across the boundaries.[5] The values of ν D are (a) 10, (b) 20, (c) 40, and (d) 60.The transmittance spectrums of the four-section HPC exhibit two types of spectral holes in the Bragg regime λ 0 ∈ (660,695) nm of a defect-free HPC.Two reflection holes in the spectrum of T RR emerge at the resonance wavelengths λ 0 1 < λ 0 2 when ν D is relatively small.As ν D increases, the two co-handed reflection holes vanish and are replaced by two transmission holes in the spectrum of T LL when ν D is sufficiently large.Also, the simultaneous occurrence of both types of spectral holes at λ 0 2 is observable in (c).hole and a cross-handed transmission hole at the same wavelength (λ 0 2 ) in the four-section HPC.The solid line marked with diamond symbols is plotted for 0 < φ t < π/2, while the dashed line marked with box symbols is plotted for π/2 < φ t < π.See Fig. 2 for other parameters.which formulas are in accord with those for single-defect HPCs [15,23].
In either of the two ν D -ranges, the simultaneous occurrence of the co-handed reflection hole and the cross-handed transmission hole at the same wavelength is impossible; although it is possible for some values of ν D that a co-handed reflection hole occurs at λ 0 2 and a crosshanded transmission hole occurs λ 0 1 , as shown in Fig. 2(b) for ν D = 20.
However, there is an exception for a tiny ν D -regime -whose center increases with increasing |π/2 − φ t | (see Fig. 3) -such that the two types of spectral holes are simultaneously present at the wavelength λ 0 2 , as shown in Fig. 2(c) for ν D = 40.The bandwidth of the concurrent spectral holes at λ 0 2 is highly localized within the wavelength-regime of the cross-handed transmission hole.
The optical response of the four-section HPC with |π/2 − φ t | >> δ φ t implies that the defect modes responsible for the generation of spectral holes are in fact coupled to different defect sites and can be excited by plane waves of different circular polarization states, depending on the value of ν D .This conclusion can be garnered from the localized distribution of energy density inside the four-section HPC, shown in Fig. 4. In this figure, the energy density for λ 0 = λ 0 2 peaks at the sites of the two φ t -twist defects -which identifies the defect mode that is coupled to the φ t -twist defect; while the energy density for λ 0 = λ 0 1 is peaked only at the site of the −φ t -twist defect -which identifies the other defect mode that is coupled to the −φ ttwist defect, regardless of the value of ν D .Therefore, these two defect modes are uncoupled to each other and effectively produce spectral holes at two different wavelengths.The two spectral holes, when highly localized in the cross-handed transmittance spectrum for sufficiently large ν D , can come very close to each other for φ t → π/2 ± .Also shown in Fig. 4 is the dependence of the excitation of the defect modes on the circular polarization state of the incident plane wave.The peak energy density for the co-handed CP planewave incidence is significantly larger (∼ 1 to 3 orders of magnitude) than that for the cross-handed CP planewave incidence for small ν D , while the relationship is reversed for sufficiently large ν D .Therefore, the two defect modes are co-handed for small ν D , but cross-handed for sufficiently large ν D ; thus, the evolution of the two types of spectral holes with the rise of ν D in Fig. 2 is reaffirmed.
However, an exceptional case is that the energy density for the co-handed incidence case can reach the peaks at the two φ t -twist defect sites that are significantly larger than the saturated peak of the energy density for the cross-handed case, as shown in Fig. 4(a 3 ) for ν D = 40.This implies that the defect mode coupled with the φ t -twist defect can be excited by both the cohanded and the cross-handed CP plane waves for the specific values of ν D ; then, the two types of spectral holes simultaneously appear at the resonance wavelength λ 0 2 , as demonstrated by Fig. 2(c).
Finally, no crossover phenomenon could be defined for the defect mode coupled to the φ ttwist defect in the four-section HPC, although the defect mode coupled to the −φ t -twist defect still qualifies for that definition.

|π/2 − φ | ≤ δ φ
The whole picture of the defect modes as discussed in Section 4.1 changes for |π/2 − φ t | ≤ δ φ t , and so does the optical response of the four-section HPC.As an example, Fig. 5 shows that when φ t = π/2, three co-handed reflection holes at different wavelengths λ 0 − < λ Br 0 < λ 0 + are present for small ν D .As ν D increases, all three reflection holes wane and are replaced by three cross-handed transmission holes.Further increase in ν D leads to all three co-handed transmission holes coalescing into one cross-handed transmission hole at λ Br 0 .Furthermore, a co-handed reflection hole and a cross-handed transmission hole were not found to be simultaneously present at a single wavelength, regardless of the value of ν D .Fig. 6.Same as Fig. 4, but for φ t = π/2 and the resonance wavelengths (a 1 -a 4 ) λ 0 = λ Br 0 and (b 1 )-(b 4 ) λ 0 = λ 0± .The defect mode coupled to the φ t -twist defect is shown in (a 1 )-(a 4 ) by the considerable peaks at the sites of the two φ t -twist defects, while the defect mode coupled to all three twist defects (or the ±φ t -twist defects) is shown in (b 1 )-(b 4 ) by the considerable peaks at the sites of all three twist defects.The two defect modes are coupled to each other.It is also noted that all three resonance wavelengths λ 0± and λ Br 0 are actually located in a single cross-handed transmission hole for sufficiently large ν D , such as ν D = 60.See Fig. 5 for other parameters.
In fact, there are still two defect modes for |π/2 − φ t | ≤ δ φ t that can be identified from the energy density distribution inside the HPC; see Fig. 6.One defect mode is always coupled to the φ t -twist defect, as evinced by the considerable peaks of the energy density at the two φ ttwist defect sites for the wavelength λ Br 0 ; see Figs. 6(a 1 )-(a 4 ).The other defect mode is always coupled to all three twist defects, as evinced by the considerable peaks of the energy density at each of three defect sites for the resonance wavelength λ 0 ± in Figs.6(b 1 )-(b 4 ).Therefore, the two defect modes are actually coupled to each other for any value of ν D .A simple explanation for that coupling is that φ t = π − φ t when φ t = π/2, so that both types of twists in the HPC are equivalent.
The defect mode coupled to the φ t -twist defect is responsible for the spectral hole at the intermediate resonance wavelength λ Br 0 , while the defect mode coupled to all three twist defects is the cause of the other two spectral holes at λ 0 ± when ν D is relatively small.The coupling of the two defect modes leads to the degeneracy of all the spectral holes for significantly large ν D such that only one cross-handed transmission hole is eventually stable at λ Br 0 , as shown in Fig. 5(d) for ν D = 60.Accordingly, the two defect modes are highly localized to merge together in the single cross-handed transmission hole when ν D is sufficiently large.

Concluding Remarks
The focus of this paper is on defect modes in a four-section HPC comprising three twist defects at the intersections, all due to just the angle φ t ∈ (0, π).Two defect modes are identified after examining the localized energy density distribution inside the four-section HPC.The characteristics of the two defect modes are summarized as follows: • One defect mode is always coupled to the φ t -twist defect; while the other defect mode is coupled to either (i) the  (1) 0 1 < λ 0 2 when ν D is relatively small.As ν D increases, all three co-handed reflection holes vanish and are replaced by two cross-handed transmission holes at the resonance wavelengths λ 0 1 < λ 0 2 when ν D is sufficiently large.Two defect modes have been identified accordingly: One is coupled to the φ t -twist defect so that it is responsible for the spectral hole at λ 0 2 ; the other is coupled to the −φ t -twist defect so that it is responsible for the spectral hole(s) at λ 0 1 (λ (1) 0 1 and λ (2) 0 1 ).Therefore, the two defect modes are uncoupled to each other.The simultaneous occurrence of the two types of spectral holes at the same resonance wavelength (λ 0 2 ) is also observable in (c).See Fig. 2   (1) 0+ when ν D is relatively small.As ν D increases, all five cohanded reflection holes wane and are replaced by five cross-handed transmission holes, and eventually they coalesce into one stable cross-handed transmission hole for sufficiently large ν D .Two defect modes have also been identified: One is coupled to the φ t -twist defect so that it is responsible for the spectral hole at λ Br 0 ; the other is coupled to both the φ t -and the −φ t -twist defects so that it is responsible for the other four spectral holes at λ • For |π/2− φ t | >> δ φ t , the two defect modes are uncoupled to each other.In consequence, two co-handed reflection holes emerge separately in the Bragg regime for small ν D , and but evolve into two stable cross-handed transmission holes for sufficiently large ν D .
• For |π/2 − φ t | ≤ δ φ t , the two defect modes are coupled to each other.The optical consequence of that coupling is that three co-handed reflection holes are present around the center of the Bragg regime for small ν D ; all three evolve into three cross-handed transmission holes as ν D increases; and eventually all three co-handed transmission holes coalesce into one stable cross-handed transmission hole for sufficiently large ν D .The two coupled defect modes merge together to manifest the single cross-handed transmission hole for sufficiently large ν D .
• When the two defect modes are uncoupled to each other, the simultaneous occurrence of a co-handed reflection hole and a cross-handed transmission hole at a single wavelength can be realized for a specific tiny ν D -regime.This means that the defect mode coupled to the φ t -twist defect can be excited by both co-handed and cross-handed CP plane waves -which is in contrast to the single CP defect mode in HPCs [11].The simultaneous occurrence of the two defect modes indicates that -at least operationally -the four-section HPC can act as if it has two structural handednesses instead of just one.Although a strict analogy cannot be drawn, we note here that equichiral layered structures have been recently found to exhibit both kinds of handednesses in their remittance spectrums [24].
The exploitation of multiple defect modes in the spectrum of a single device can boost the efficiency of optical transmission systems.The two defect modes discussed for the four-section HPC apply in general to any 2 -section ( ≥ 2) HPC with 2 − 1 intersectional twist defects of ϕ j+1 − ϕ j = (−1) j+1 φ t for 1 ≤ j ≤ 2 − 1.But in particular, the number of spectral holes increases with the rise of for small thickness, although the cross-handed transmission hole, when it becomes stable for sufficiently large thickness, can only be located at one or two resonance wavelengths, depending on whether the two defect modes are coupled or not; see Figs. 7 and 8 as examples for = 3.In contrast, on applying the CWT presented in Section 3, it follows that a three-section or five-section HPC will not exhibit such defect modes stably (i.e., their occurrence is independent of the sectional thickness); and this conclusion can be generalized to (2 − 1)-section ( ≥ 2) HPCs.

Fig. 3 .
Fig. 3.The relation of ν D to |π/2 − φ t | for the simultaneous occurrence of a co-handed reflection

Fig. 4 .
Fig. 4. Distribution of the normalized energy density of the electromagnetic fields inside the foursection HPC with φ t = π/3, computed for the resonance wavelengths (a 1 )-(a 4 ) λ 0 = λ 0 2 and (b 1 )-(b 4 ) λ 0 = λ 0 1 .The solid lines are for co-handed CP planewave incidence, while the dotted lines are for cross-handed CP planewave incidence.The values of ν D are (a 1 , b 1 ) 10, (a 2 , b 2 ) 20, (a 3 , b 3 ) 40, and (a 4 , b 4 ) 60.The defect mode coupled to the φ t -twist defect is evinced in (a 1 )-(a 4 ) by the peaks at the sites of the two φ t -twist defects, while the defect mode coupled to the −φ t -twist defect is evinced in (b 1 )-(b 4 ) by the peak at the site of the −φ t -twist defect.The two defect modes are uncoupled to each other.See Fig.2for other parameters.

WavelengthFig. 8 .
Fig.8.Same as Fig.7, but for φ t = π/2.Five co-handed reflection holes emerge at the resonance wavelengths λ , the two defect modes are coupled to each other, which leads to the degeneracy of all the spectral holes for sufficiently large ν D .It is also true that the two defect modes are actually localized in a single cross-handed transmission hole for sufficiently large ν D , such as ν D = 50.(C) 2005 OSA 19 September 2005 / Vol. 13, No. 19 / OPTICS EXPRESS 7334 defects for |π/2 − φ t | ≤ δ φ t , where δ φ t > 0 is a very small angle for sufficiently large ν D .

Table 1 .
The value of δ φ t for various values of ν D calculated by (41).See Fig.2for other parameters.