1. 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 [2–4]. Analogously, in photonics, unidirectional edge modes can be also implemented using photonic crystals (PhCs) with static magneto-optical effects [5–12] or dynamic phase modulations (including spatially-equivalent ones) [13–15].

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 [16–18]. 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 [20–25]. 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) [26–32]. 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.

2. 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.

 

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.

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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 $\text{exp}(im\varphi )/\sqrt{2\pi }$ 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 left-handed circularly-polarized (LHCP) and right-handed circularly-polarized (RHCP) waves with polarizations ${\widehat{e}}_{+}=\left(\widehat{x}+i\widehat{y}\right)/\sqrt{2}$ and ${\widehat{e}}_{-}={\widehat{e}}_{+}^{*}$, 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_zP) 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 LHCP-like 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.

 

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.

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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 cross-sectional 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.

3. 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.

3.1. 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 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:

3.2. Generalized reciprocity theorem in waveguide-like structures and mode orthogonality

Let us consider two sets of electric and magnetic fields [E₁(r), H₁(r)] and [E₂(r), H₂(r)] which are independently generated by two sources J₁(r) and J₂(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]:

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 [ ${\mathbf{E}}_{mns}^{\sigma }(\mathbf{r})$, ${\mathbf{H}}_{mns}^{\sigma }(\mathbf{r})$] of these modes as

3.3. 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 $\left[{\mathbf{E}}_1(\mathbf{r}),{\mathbf{H}}_1(\mathbf{r})\right]=\left[{\mathbf{E}}_{-m,ns}^{-\sigma }(\mathbf{r}),{\mathbf{H}}_{-m,ns}^{-\sigma }(\mathbf{r})\right]$ and [E₂(r), H₂(r)] = [E_s(r), H_s(r)]. In this way, equation (3) is rewritten as

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:

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 [ $A_{mns}^{\sigma }(z)$, $J_{mns}^{\sigma }(z)$] 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 [ $A_m^{\sigma }(z)$, $J_m^{\sigma }(z)$], ${\mathit{ℰ}}_m^{\sigma }(\mathit{\rho })$, 𝒩_m(ω), and ${\kappa }_{m{m}^{\prime }}^{\sigma {\sigma }^{\prime }}(\omega )$, 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 $A_m^{\sigma }(z)$ behaves like the phase factor exp(iσqz). In views of this behavior, we define a slowly-varying amplitude ${\tilde{A}}_m^{\sigma }(z)$ and its source counterpart ${\tilde{J}}_m^{\sigma }(z)$ in addition to the rapid spatial variation of exp(iσqz) as

With the aforementioned approximations, we rewrite Eq. (11a) using the four slowly-varying pairs [ ${\tilde{A}}_m^{\sigma }(z)$, ${\tilde{J}}_m^{\sigma }(z)$] in a 4-by-4 matrix form of differential equations as

The matrix M is block diagonal. Only its second and third rows/columns are coupled. The eigenvalues λ_u(ω) (u = 1 to 4) of M can be obtained from its characteristic polynomial and are

In the chiral bandgap, only the LHCP-like modes $A_{+}^{\text{f}}(z)$ and $A_{-}^{\text{b}}(z)$ 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 forward-propagating LHCP-like mode with a propagation constant q + λ₁(ω) = β(ω) − κ⁽⁰⁾(ω):

4. Comparison between theory and numerical results

4.1. 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′+[β(ω)−κ⁽⁰⁾(ω)]z′})≈exp[2i(ϕ′+qz′)], and the second one is the transverse mode profile ${\mathit{ℰ}}_{+}^{\text{f}}({\mathit{\rho }}^{\prime })$.

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

For effects of ${\mathit{ℰ}}_{+}^{\text{f}}({\mathit{\rho }}^{\prime })$, 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 ${ℰ}_{\pm ,\rho }^{\text{f}}({\rho }^{\prime })$ and ${ℰ}_{\pm ,\varphi }^{\text{f}}({\rho }^{\prime })$ of the fundamental TE mode profiles ${\mathit{ℰ}}_{\pm }^{\text{f}}({\mathit{\rho }}^{\prime })$ have radial dependencies as follows (dropped prefactors are set real):

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 $A_{+}^{\text{b}}(z)$, which is generated by the source $J_{+}^{\text{b}}(z)$ and related to an integrand $\mathrm{\Delta }{\epsilon }_{\text{r},\text{s}}({\mathbf{r}}^{\prime },\omega ){\mathit{ℰ}}_{-}^{\text{f}}({\mathit{\rho }}^{\prime })\cdot {\mathit{ℰ}}_{-}^{\text{f}}({\mathit{\rho }}^{\prime })\propto {\left|{\mathit{ℰ}}_{+}^{\text{f}}({\mathit{\rho }}^{\prime })\right|}^2\propto f_{+}({\rho }^{\prime })$, 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.

 

Fig. 3 The two functions f_∓(ρ_s) versus ρ_s. 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.

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4.2. 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_sW_s = 0.3πR². 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 first-order 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 computational accuracies and dense meshes used in calculations. However, it is certain that the smallness of |r(ω)|² originates from the inability of 2-fold RH scatters to compensate the phase factor exp(2i{ϕ′+[β(ω)−κ⁽⁰⁾(ω)]z′}) in Eq. (19). In Fig. 4(d), we show the field distributions |E(r)|² around the 2-fold LH and RH scatters with R_s = 0.7R and Δε_r,s = 0.75 on the y − z plane. The RH one seems to backscatter little, as can be told from seemingly unaltered magnitudes of the propagating waves at its two sides.

 

Fig. 4 (a) Cross sections of 2-fold LH scatters as R_s is varied. The cross-sectional areas are fixed at 0.3πR². (b) The reflectivities of the 2-fold LH scatters as a function of R_s at various Δε_r,s. The maximal backscattering occurs at the largest R_s in these calculations. (c) The reflectivities of the 2-fold RH scatters as a function of R_s at various Δε_r,s. They are much smaller than their counterparts in (b). (d) The field distributions near the 2-fold LH (top) and RH (bottom) scatters with R_s = 0.7R and Δε_r,s = 0.75 on the y − z plane, respectively. The incident field is hardly backscattered by the RH scatter.

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Similar to the 2-fold RH scatter, the 3-fold and 4-fold LH scatters do not backscatter significantly because they only have the azimuthal fourier components at |l| ≥ 3 and hence cannot compensate the phase factor. The power reflectivities |r(ω)|² of these two scatters as a function R_s at different Δε_r,s are shown in Fig. 5(a) and (b), respectively. The field distributions near the 3-fold and 4-fold LH scatters with R_s = 0.3R and Δε_r,s = 0.75 are depicted in Fig. 5(c). There are significant local fields confined in the two scatters. However, the local fields mostly correspond to the evanescent modes in the chiral PhC. One can tell that the magnitudes of the propagating waves in the front and backsides of the two scatters remain nearly identical.

 

Fig. 5 (a) The reflectivities of the 3-fold LH scatters versus R_s at various Δε_r,s, and (b) the counterparts of the 4-fold LH scatters. Both scatters do not backscatter significantly. (c) The field distributions near the 3-fold (left) and 4-fold (right) LH scatters with R_s = 0.3R and Δε_r,s = 0.75 on the y − z plane, respectively. Except for the prominent local fields, the magnitudes of the propagating waves change little in these two cases.

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The expression of the reflection coefficient r(ω) in Eq. (22) indicates that the rotationally asymmetric LH scatters (g = 1) can also backscatter as long as the azimuthal Fourier component at l = −2 is present. From this point, we pick up a group of such scatters exemplified by the inset in Fig. 6(a) intentionally. These LH scatters have no azimuthal Fourier components at l = −2 and should not bring about the prominent backscattering in theory. However, their reflectivities at Δε_r,s = 0.51, 0.64, and 0.75 are quite significant, as shown in Fig. 6(a). In addition, the dependencies of the reflectivities on R_s are similar to the function |f₊(R_s)|². These phenomena mark the breakdown of the first-order Born approximation. We suggest that the failure may be caused by the too large permittivity variation Δε_r,s. As shown in Fig. 6(b), we choose two scatter radii R_s = 0.27R and 0.7R and numerically calculate |r(ω)|² as a function of Δε_r,s. Indeed, at R_s = 0.27R, no significant reflections occur at Δε_r,s ≤ 0.2, which indicates that the first-order Born approximation is still valid in this regime. The working range of Δε_r,s for the first-order Born approximation is even larger at R_s = 0.7R. From these observations, we speculate that the validity of the first-order Born approximation could be quite limited for the chiral scatters with asymmetric cross sections as compared to those with high-order rotational symmetries.

 

Fig. 6 (a) The reflectivities of the rotationally asymmetric LH scatters versus R_s at various Δε_r,s. While the scatters have no azimuthal Fourier components at l = −2, the backscattering is prominent. (b) The reflectivities of the rotationally asymmetric LH scatters with R_s = 0.27R and 0.7R versus Δε_r,s, respectively. The first-order Born approximation breaks down as Δε_r,s becomes too large.

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At last, we model the numerical reflection coefficient of a small scatter as a function of its position ρ_s from the center of the chiral PhC using the functional form of f₋(ρ_s). As shown in the inset of Fig. 7, we consider a small copper block with a size of 0.3R × 0.5R × 0.3R and move it along the direction perpendicular to the line joining the centers of the circular WG and bump. Since the true LHCP-like propagating modes in the chiral PhC slightly deviate from the guiding modes of the circular PEC WG, and the copper block is not infinitesimally small, a finite residual reflection is expected at ρ_s = 0 even though the complete transparency is suggested by f₋(ρ_s = 0). Therefore, we approximate the realistic reflection coefficient r(ω) of the copper scatter with the following form:

 

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.

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5. 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 PhC 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.