immunity to backscattering in chiral one-way photonic crystals

We show that the propagating modes in a strongly-guided chiral one-way photonic crystal are not backscattering-immune even though they are indeed insensitive to many kinds of scatters. Since these modes are not protected by the nonreciprocity, the backscattering does occur under certain circumstances. We use a perturbative method to derive criteria for the prominent backscattering in such chiral structures. From both our theory and numerical examinations, we find that the amount of backscattering critically depends on the symmetry of scatters. Additionally, for these chiral photonic modes, disturbances at the most intense parts of field profiles do not necessarily lead to the most effective backscattering. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (290.1350) Backscattering; (290.5855) Scattering, polarization. References and links 1. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. 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Introduction
The reduction of unwanted reflections and backscattering during wave transmissions and propagations is important in photonic integrated systems.It lowers potential channel noises and interferences and therefore necessitates the device isolator [1].Such devices only transmit the optical power along one direction but block the energy flow in the other.To implement this functionality in a robust manner, nonreciprocal setups are required.In electronics, the one-way phenomenon is present in the edge states of quantum Hall effect due to a high magnetic field that breaks the time-reversal symmetry [2][3][4].Analogously, in photonics, unidirectional edge modes can be also implemented using photonic crystals (PhCs) with static magneto-optical effects [5][6][7][8][9][10][11][12] or dynamic phase modulations (including spatially-equivalent ones) [13][14][15].
Despite various nonreciprocal schemes aimed at one-way behaviors of waves, their compatibilities with photonic systems nowadays still require some key progress.Since most of the conventional linear optical devices are reciprocal, great efforts have to be devoted to the integrations of nonreciprocal (magnetic) materials or complicated time-variant controls to photonic structures.These inconveniences motivate the development of reciprocal schemes which distinguish or isolate energy flows in opposite directions.For example, asymmetric two-way transmissions may be achieved through designed gratings [16][17][18].In wave-guiding structures, reciprocal modes that are backscattering-immune to a variety of scatters are preferred so that energy flows are not easily reversed.This, however, does not mean only unidirectional modes are supported.In fact, in reciprocal environments, counter-propagating modes always exist simultaneously, namely, a reciprocal photonic system is bidirectional [19].
Two factors are essential to the immunity of modes (states) to backscattering in reciprocal (time-reversible) systems.As long as (i) the counter-propagating modes (states) are firmly associated with some orthogonal degrees of freedom (DOFs), and (ii) elastic scatters do not mix these DOFs, the backscattering is suppressed.Such characteristics can be found in the edge modes (states) of photonic (electronic) topological insulators [20][21][22][23][24][25].In fact, these prerequisites on one-way propagations are also present in circularly-polarized (CP) guided modes of one-dimensional (1D) chiral PhCs or waveguides (WGs) [26][27][28][29][30][31][32].In this case, the key DOFs are the two circular polarizations rotating in opposite orientations.Absence of the backscattering from simple scatters has also been demonstrated experimentally in 1D chiral PhCs [31].
While the aforementioned condition (ii) may hold for a broad range of scatters, there are always exceptions.The failure of this condition marks the onset of backscattering.This point motivates us to examine the robustness of reciprocal chiral guided modes against the backscattering.In this work, we look into the dependencies of backscattering on geometries of different scatters in a 1D chiral PhC covered by perfect electric conductors (PECs) at microwave frequencies.We use the first-order Born approximation and coupled-mode theory to develop the criteria of prominent scattering in the chiral structure.The outcomes indicate that the amount of backscattering closely depends on the cross-sectional symmetry of scatters.In addition, even if the scatter is placed at positions corresponding to the most intense parts of mode profiles, the backscattering there is not necessarily the most prominent.The behavior is contrary to that of typical backscattering in generic PEC WGs.All these features are qualitatively verified by rigorous numerical calculations.Our studies also point out what types of scatters or defects should be avoided in one-way applications of chiral structures so that the backscattering could be minimized.While this work is aimed at microwave frequencies, the results may be generalized to analogous plasmonic chiral PhCs at optical frequencies after the structure size and operating wavelength are properly scaled down.
The rest of this paper is organized as follows.In section 2, the structure and modes of the 1D chiral PhC as well as the setup of backscattering will be briefly introduced.The theoretical model based on coupled-mode theory are outlined in section 3.In section 4, we will discuss the inferences from our theory on backscattering and present the numerical results and their comparisons with theoretical predictions.The conclusion will be given in section 5.

Structure of chiral one-way photonic cyrstal
The schematic diagram of the 1D chiral PhC in this study is shown in Fig. 1.This chiral structure is generalized from a circular PEC WG with its radius R = 1 cm and center coincident with the z axis.The interior of the PhC is filled with air and has a relative permittivity of unity.A generic cross section of the PhC is shown in the top inset of the same figure.A PEC bump in the form of a truncated sector is present at the circular rim of the cross section.The thickness ΔR and outer arc length ΔS of the bump are 0.15R and 0.3R, respectively.The chiral PhC is a right-handed (RH) structure and has a pitch P. Since its cross section does not have any symmetries, the pitch P is also the period of this PhC.The bottom inset of Fig. 1 shows a scatter to be inserted into the PhC.We focus on chiral scatters with a pitch P s and cross sections of different rotational symmetries.As will be shown later, chiral scatters reflect the propagating modes effectively even though other types of scatters could also result in backscattering.
In absence of the bump, eigenmodes of the circular PEC WG are analytically solvable [33].They can be divided into transverse-electric (TE) modes TE mn and transverse-magnetic (TM) modes TM mn , where m ∈ Z and n ≥ 1 are the azimuthal and radial mode numbers, respectively.Hereafter, we adopt the representation exp(imφ )/ √ 2π for azimuthal parts of various cylindrical field components, where φ is the azimuthal angle.The fundamental modes are two degenerate TE modes TE ±1,1 .These two modes are of interest since their polarization patterns resemble circular polarizations and can properly grasp features of the chirality.With optic conventions for waves propagating toward the +ẑ direction, the TE +1,1 and TE −1,1 modes are similar to the lefthanded circularly-polarized (LHCP) and right-handed circularly-polarized (RHCP) waves with polarizations ê+ = ( x + i ŷ)/ √ 2 and ê− = ê * + , respectively.On the other hand, upon reversing the propagation toward the −ẑ direction, the LHCP and RHCP waves change their polarizations into ê− and ê+ and are therefore associated with TE −1,1 and TE +1,1 modes, respectively.
To make the chiral nature of backscattering clear, we attempt to rule out the characteristics of higher-order modes other than TE ±1,1 ones in this process.Therefore, we adopt the frequency range below which the next guided mode TM 0,1 and other even higher-order modes are all cut off.Accordingly, the pitch P is set to 6 cm so that the chiral bandgap responsible for one-way propagations is opened between the cutoff frequencies of TE ±1,1 and TM 0,1 modes.Figure 2(a) shows bandstructures of the chiral PhC around the frequency range of interest in the first Brillouin zone (BZ) for wave number k z ≥ 0. The calculations are carried out with the eigenfrequency solver of commercial software COMSOL.A phase difference exp (ik z P) is imposed on the fields at two sides of the unit cell to setup the boundary condition of Bloch modes.The dispersion curves of TE ±1,1 modes are also shown in the scheme of reduced zones for comparisons.The chiral bump breaks the degeneracy of TE ±1,1 modes and turn them into the LHCPlike and RHCP-like modes.While the dispersion curve of LHCP-like modes is only slightly shifted in frequencies as compared to that of TE ±1,1 modes, an additional chiral bandgap from 9.90 to 10.42 GHz is developed on the counterpart of RHCP-like modes at the BZ center due to the RH chiral structure.In the frequency range of this chiral bandgap, only LHCP-like modes can propagate while RHCP-like ones behave like evanescent waves.Since the forward-and backward-propagating LHCP-like modes have distinct polarizations patterns similar to ê+ and ê− , respectively, the two waves would simply pass by those scatters which cannot mix polarizations ê± and hence are not effectively reflected.
As a test of the similarity between the TE +1,1 and forward-propagating LHCP-like modes, we monitor the incidence of the TE +1,1 mode from the circular WG to the chiral PhC at a frequency of 10.25 GHz within the chiral bandgap, which is also the frequency of LHCP-like modes at the BZ center.The cross-sectional view of the field pattern near the junction of the WG and PhC is illustrated in Fig. 2(b).Only minor reflections and field variations due to the discontinuity of the wave-guiding structure are observed.As shown at the bottom of Fig. 2(b), the crosssectional distributions of field strengths in the WG and PhC regions are similar except for areas near the bump.This examination confirms the resemblance between the TE +1,1 mode of the circular WG and forward-propagating LHCP-like mode of the chiral PhC.Since no significant reflections are observed here, this incidence scheme will also be utilized in later calculations to excite the forward-propagating LHCP-like mode inside the chiral PhC.

Theoretical model of backscattering
We use a perturbative scheme to investigate the backscattering in chiral structures.While the chiral PEC bump in Fig. 1 and some scatters to be considered are not really weak perturbations, the derivation here does provide some insights into the backscattering in this chiral PhC.In sections 3.1 and 3.2, we briefly describe some preliminary knowledge on the modeling.Details about the backscattering are presented in sections 3.3.

Scatters as effective radiation sources: first-order Born approximation
In the presence of a scatter, the total field E(r) in a chiral PhC, of which the permittivity profile is slightly different from that of a circular PEC WG, satisfies the following wave equation: the scatter and is our target.If the scatter is a mild one, we may use the perturbation theory [in the order of Δε r,s (r, ω)] to approximate the governing equation of E s (r).
The zeroth-order equation is the wave equation of E i (r) in the scatter-free chiral PhC.Along with the first-order equation of E s (r), they are shown as follows: where J s (r) is an effective source resulted from the coupling between E i (r) and Δε r,s (r, ω).Equation (2b) indicates that the scattered field E s (r) could be regarded as being radiated from the source J s (r) in the chiral PhC.Such a simplification is analogous to the first-order Born approximation in quantum mechanics [34].It turns the full calculation of E(r) into an effective radiation problem, which could be further analyzed through mode expansions later.

Generalized reciprocity theorem in waveguide-like structures and mode orthogonality
Let us consider two sets of electric and magnetic fields [E 1 (r), H 1 (r)] and [E 2 (r), H 2 (r)] which are independently generated by two sources J 1 (r) and J 2 (r) at a frequency ω in two different WG-like structures 1 and 2 extending along the z axis, respectively.The permittivity distributions of structures 1 and 2 are denoted as ε r,1 (r, ω) and ε r,2 (r, ω), and their relative permeabilities μ r,1 and μ r,2 are both assumed to be unity.At a common transverse cross section A of the two structures at z, which contains areas of nonvanishing fields, the two sets of fields satisfy the following generalized reciprocity theorem in the differential form [35]: Later, we will set [E 1 (r), H 1 (r)] to modal fields of the circular PEC WG with J 1 (r) = 0 and ε r,1 (r, ω) = ε b (ω), ∀ 0 < ρ < R. The fields [E 2 (r), H 2 (r)] will be identified as the scattered fields [E s (r), H s (r)] generated by J 2 (r) = J s (r) in the chiral PhC [ε r,2 (r, ω) = ε r,c (r, ω)], where H s (r) is the magnetic field associated with E s (r).
As an application of Eq. ( 3), we obtain the orthogonality relation for modes in the circular PEC WG.Let us denote the electric and magnetic fields [E σ mns (r), H σ mns (r)] of these modes as where s indicates TE/TM types; σ = +1 and −1 (or "f" and "b") means forward-and backwardpropagating/evanescent modes along the z axis, respectively; E E E σ mns (ρ ρ ρ) and H H H σ mns (ρ ρ ρ) are transverse profiles of mode (m, n, s, σ ) aside from the azimuthal part exp(imφ )/ √ 2π; and β mns is the associated propagation constant which is identical to β −m,ns (degenerate if m = 0) and is either positive (propagating) or purely imaginary with Im[β mns ] > 0 (evanescent).We then assign two sets of fields in Eq. (3) as follows: where (m , n , s , σ ) are mode labels for the second set of fields.With ε r,1 (r, ω) = ε r,2 (r, ω) and J 1,2 (r, ω) = 0, the right-hand side (RHS) of Eq. ( 3) vanishes.Hence, the left-hand side (LHS) of the same equation, after dropping the factor exp[i(σβ mns + σ β m n s )z], has to be zero: where the area A is now the circular cross section of the WG.In Eq. ( 6), if σβ mns +σ β m n s = 0, the surface integral can be nonzero.This situation occurs only when (i) σ = −σ (counterpropagating/evanescent modes), and (ii) β mns = β m n s (degenerate modes).Condition (ii) is further limited to the case m = −m since the integration of exp[i(m + m )φ ]/2π over φ in Eq. ( 6) brings about a Kronecker delta δ m+m ,0 .These conditions lead to the following mode orthogonality relation in the circular PEC WG: where N mns (ω) is a (complex) normalization factor which could be set frequency-dependent but must be symmetric with respect to m and −m, namely, N mns (ω) = N −m,ns (ω).

Coupled-mode theory
We utilize Eq. ( 3) to formulate the coupled-mode theory for scattered fields E s (r) and H s (r) in the chiral PhC.As mentioned in section 3.2, we may set In this way, Eq. ( 3) is rewritten as in which the chiral component Δε r,c (r, ω) of the permittivity profile ε r,c (r, ω) inside the chiral PhC appears at the RHS.This chiral component Δε r,c (r, ω) can be reexpressed as r,c (ρ, ω)e il(φ −qz) , ( where q = 2π/P is the reciprocal period of pitch P; and Δε (l) r,c (ρ, ω) is the azimuthal Fourier component of Δε r,c (r, ω)| z=0 at order l.The chiral PhC is a RH (LH) structure if the argument φ − qz (φ + qz) is adopted for Δε r,c (r, ω).Therefore, we use φ − qz for the structure in Fig. 1.
The scattered fields E s (r) and H s (r) could be expanded well, though incompletely, with transverse mode profiles in the circular PEC WG: where A σ m n s (z) is the z-dependent amplitude of mode (m , n , s , σ ).The amplitude A σ m n s (z) contains the characteristics of Bloch modes in the 1D chiral PhC and features of the scatter [source J s (r)].We then substitute Eq. (10) into Eq.( 8).The surface integral at the LHS of Eq. ( 8) directly picks up the amplitude A σ mns (z) through the orthogonality relation in Eq. ( 7) and leads to the following differential equation: where κ σ ,σ (mns),(m n s ) (ω) are the coupling coefficients between various modal amplitudes; J σ mns (z) is the projection of the source J s (r) to mode (m, n, s, σ ); and we have used the integral identity Equation (11a) could be significantly simplified after some physical approximations.As discussed in section 2, the frequency range of the chiral bandgap is designed such that only the TE ±1,1 modes of the original circular PEC WG are not cut off.Therefore, we solely consider the four relevant amplitude-source pairs [A σ mns (z), J σ mns (z)] related to the fundamental TE modes with (m, σ ) = (±1, f/b).For simplicity, we abbreviate these four pairs, their associated transverse electric-field profiles, normalization factors, and coupling coefficients as , and κ σσ mm (ω), respectively, by dropping mode labels n = 1 and s = TE.Also, the identical propagation constants of these four modes are simply denoted as β (ω).
Around the frequency range of the chiral bandgap, the propagation constant β (ω) is close to q = 2π/P.Therefore, the amplitude A σ m (z) behaves like the phase factor exp(iσ qz).In views of this behavior, we define a slowly-varying amplitude Ãσ m (z) and its source counterpart Jσ m (z) in addition to the rapid spatial variation of exp(iσ qz) as Equation ( 12) also suggests that the phase difference between two amplitudes A σ m (z) and A σ m (z) is approximately reflected on the factor exp[i(σ − σ )qz].Comparing this factor with the counterpart exp[−i(m − m )qz] at the RHS of Eq. (11a), we match their phase variations and, hence, merely keep the coupling coefficients κ σσ mm (ω) with σ − σ = −(m − m ).This simplification is the major concept of the coupled-mode theory.
In the chiral bandgap, only the LHCP-like modes A f + (z) and A b − (z) corresponding to eigenvalues λ 1,4 (ω) can propagate.Their magnitudes directly determine the amounts of power flows radiated by J s (r) in the forward and backward directions.Let us consider a scatter which is present inside the chiral PhC at z ∈ [z L , z R ], where z L and z R are the leftmost and rightmost ends of the scatter.The incident field E i (r) is assumed to be solely composed of the forwardpropagating LHCP-like mode with a propagation constant q + λ 1 (ω) = β (ω) − κ (0) (ω): where A f +,i is the amplitude of the incident field; and the reference point is now set at z L .From Eqs. (2b), (11c), and (15), the projected source J b − (z) corresponding to the backwardpropagating LHCP-like mode is We may directly solve With the boundary condition We then define the reflection coefficient r(ω) as the ratio between the backward-propagating amplitude A b − (z L ) and incident amplitude A f +,i (z L ) = A f +,i to quantify the amount of backscattering.Substituting Eq. ( 16) into Eq.( 18), we express r(ω) as where Ω s is the scatter region.From Eq. ( 19), the backscattering in the chiral PhC depends on a volume integral involving Δε r,s (r, ω).This dependency will be addressed in the next section.

Criteria for prominent backscattering
The reflection coefficient r(ω) in Eq. ( 19) is affected by the interplay between Δε r,s (r, ω) and two factors.The first factor is the phase part exp(2i{φ and the second one is the transverse mode profile E E E f + (ρ ρ ρ ).For the former, the cancelation of exp[2i(φ + qz )] generally enhances the volume integral in Eq. ( 19) even though the effect is relatively minor for small scatters.In other words, the prominent backscattering would occur if Δε r,s (r , ω) ∼ exp[−2i(φ + qz )] in the frequency range of the chiral bandgap.From the analogy of the chiral component Δε r,c (r, ω) in Eq. ( 9), we expect an effective scatter to be a LH chiral structure with (i) a pitch P s close to P, and (ii) a significant azimuthal Fourier component of Δε r,s (r , ω) at the order l = −2.Let us further write the permittivity variation Δε r,s (r , ω) of such a LH chiral scatter as r,s (ρ , ω)e il(φ +q s z ) , (20) where U(r ) is an indicator function which is unity in Ω s but zero elsewhere; and q s = 2π/P s .
If Δε r,s (r , ω) has g-fold rotational symmetry along the z axis (invariant if φ → φ + 2π/g), only the components Δε (l=τg) r,s (ρ , ω) (τ ∈ Z) are present, namely, the components exist only if their orders l are multiples of g.It can be verified that for g ≥ 3, the component Δε r,s (ρ , ω) vanishes.Therefore, the prominent backscattering only occurs for proper LH chiral scatters with two-fold or rotationally asymmetric cross sections (g = 1, 2).
For effects of E E E f + (ρ ρ ρ ), we consider a small scatter region Ω s with ρ − ρ s ∈ [−Δρ s /2, Δρ s /2], where ρ s is the transverse distance of the scatter, and Δρ s is its range (Δρ s /R 1).The radial and azimuthal components E f ±,ρ (ρ ) and E f ±,φ (ρ ) of the fundamental TE mode profiles E E E f ± (ρ ρ ρ ) have radial dependencies as follows (dropped prefactors are set real): where J 1 and J 1 are Bessel function of the first kind and its derivative at the order 1, respectively; and k t ≈ 1.84/R is the transverse wave number of the fundamental TE modes.We note that and rewrite r(ω) in Eq. (19) as where "• • •" represents details other than the radial integral; ρ s Δρ s is the scatter area per unit azimuthal angle at ρ = ρ s ; Δε r,s (r , ω) ρ =ρ s is an average of Δε r,s (r , ω) at ρ = ρ s ; f − (ρ s ) is a function characterizing the position dependency of small scatters in the chiral PhC; and f + (ρ s ) describes the radial dependency of |E E E f + (ρ ρ ρ )| 2 ρ =ρ s .The two functions f ∓ (ρ s ) are depicted in Fig. 3 for comparisons.The function f − (ρ s ) vanishes at ρ s = 0 but peaks at ρ s ≈ 0.81R, while f + (ρ s ) is the largest and smallest at ρ s = 0 and R, respectively.The behavior of f − (ρ s ) indicates that the prominent backscattering in the chiral PhC takes place as the scatter is near the rim of the structure, at which the modal intensity is low.The reason for the oddness is that the scattering channels from LHCP-like to RHCP-like modes are suppressed, as the latter become evanescent within the chiral bandgap.If the chirality of the PhC were absent, the backscattered amplitude A b + (z), which is generated by the source J b + (z) and related to an integrand Δε r,s (r , ω) , could carry power flows.This backscattered amplitude would then show the typical scattering trend of f + (ρ s ) for the fundamental TE modes in circular PEC WGs.

Numerical results
With the LHCP-like incident mode of the circular PEC WG at 10.25 GHz, we solve the scattered field inside the chiral PhC using the three-dimensional finite-element method implemented in COMSOL.The two ends of the computation domain are set to perfectly matched layers to make the scattered field an outgoing wave.The power reflectivity |r(ω)| 2 is then obtained from the incident and total fields.Unless otherwise mentioned, there are 32 pitches of chiral structures in the front and backside of scatters.We first consider fictitious chiral scatters with g-fold (g = 1 to 4) rotationally (a)symmetric cross sections, which are composed of 2g identical truncated sectors with permittivity variations alternating in signs [Δε r,s (r, ω) = ±Δε r,s ].For conveniences, we will simply call the scatters as g-fold LH/RH scatters hereafter (g ≥ 2).The radii R s of these scatters will be varied, but for fair comparisons, their radial widths W s shall be adjusted accordingly so that the cross-sectional areas A s remain unaltered.
The cross-sectional geometry of the 2-fold LH scatters as the radius R s increases is shown in Fig. 4(a) (arrow indicates the handness).The cross-sectional areas are fixed at A s = 2πR s W s = 0.3πR 2 .The pitches P s and lengths of the scatters are both set to P. The reflectivities |r(ω)| 2 versus R s at different permittivity variations Δε r,s = 0.51, 0.64, and 0.75 are shown in Fig. 4(b).Although these variations are comparable to the background permittivity ε r,b (ω) = 1, the trends of |r(ω)| 2 versus R s qualitatively follow the function | f − (R s )| 2 in Fig. 3, indicating that the firstorder Born approximation and coupled-mode theory work well in these cases.As R s ≈ 0.27R, the scatter cross sections are filled circles and have the best overlap with the most intense portion of the LHCP-like mode.However, the corresponding power reflectivities are minimal.As R s becomes large, the reflectivities significantly increase.This characteristic indicates that to avoid the prominent backscattering in this chiral PhC, the fluctuations and defects near the rim of the wave-guiding structure should be reduced.On the other hand, the 2-fold RH scatters only backscatter weakly, as shown in Fig. 4(c).They exhibit small power reflectivities that do not seem to have a consistent trend as R s increases, partly due to the compromise between where a is a proportional factor; and b accounts for the residual reflection.In numerical calculation, only 8 pitches of the chiral structures are placed in the front and backside of the scatter due to the necessary fine meshes around the copper block.The numerical data are then fitted with |r(ω)| 2 based on Eq. ( 23).As shown in Fig. 7, the fitting curve using Eq. ( 23) agrees well with the numerical data.The results also indicate that an arbitrary scatter, even if it is not chiral, also backscatters more efficiently as it is farther away from the center of the chiral PhC.

Conclusion
We have analyzed the backscattering in a chiral one-way PhC at microwave frequencies.A perturbative method is utilized to obtain criteria for the prominent backscattering.The scattered amplitude depends on the azimuthal Fourier components of scatter cross sections at order |l| = 2. Chiral scatters without these Fourier components would not reflect the chiral propagating modes efficiently.This leads to the close dependency of the backscattering on rotational symmetries of chiral scatters.In addition, the disturbance at high-intensity points of the chiral mode does not necessarily lead to the most effective backscattering.Scatters near the rim of the actually backscatter more easily.These characteristics are all qualitatively verified with numerical calculations of the backscattering based on different configurations of scatterers.They also reveal what types of scatters or defects should be avoided in one-way applications of chiral structures in order to minimize the backscattering.

Fig. 1 .
Fig. 1.The schematic diagram of the one-way chiral PhC covered by PECs.The top inset shows a generic cross section inside the PhC.The bottom inset indicates a chiral scatter to be inserted into the the wave-guiding structure.

Fig. 2 .
Fig. 2. (a) Bandstructures of the 1D chiral PhC.The dispersion curves of unperturbed TE ±1,1 modes are split into those of the LHCP-like and RHCP-like modes.The RHCP-like mode has a chiral bandgap at the BZ center.(b) The incidence of a forward-propagating TE +1,1 mode from the circular WG into chiral PhC.The cross-sectional distributions of square field magnitudes in the WG and PhC regions are shown at the bottom.Except for areas near the bump, the two field patterns look similar.

Fig. 7 .
Fig.7.The reflectivity of a small copper block with a size of 0.3R × 0.5R × 0.3R versus the position ρ s .The fitting curve shows a decent agreement with the numerical data.