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Effective medium theory with closed-form expressions for bi-anisotropic optical metamaterials

Neng Wang and Guo Ping Wang

Open Access Open Access

Abstract

Bi-anisotropic optical metamaterials are playing an increasingly important role in current wave-functional metamaterials and topological photonics due to their extra degree of freedom in addition to the permittivity and permeability. In this work, we derived the closed-form expressions for effective constitutive parameters of 2-dimensional (2D) bi-anisotropic metamaterials whose chirality tensors can possess both diagonal and off-diagonal components in the long-wavelength limit based on the Mie theory. Our formulas can be regarded as an extension of the Maxwell-Garnet formula to 2D bi-anisotriopic metamaterials and are verified through full wave numerical simulations. These closed-form formulas will benefit the design and analysis of the optical properties of 2D bi-anisotriopic metamaterials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recently, a great deal of attention has been paid to bi-anisotropic metamaterials due to their intriguing properties in wave manipulations, such as negative refraction [1–4], strong optical activity [5–7] and circular dichroism [8–10]. Bi-anisotropic metamaterials have also been widely noticed as a fruitful platform for realizing nontrivial topological phenomena in photonics [11–14], due to the inversion symmetry breaking and strong spin-orbital coupling effects.

In the framework of effective medium theory (EMT) [15], the scattering properties of metamaterials as well as the propagating properties of light inside metamaterials are sufficiently described by the effective constitutive parameters. As a result, acquiring these effective constitutive parameters correctly and efficiently is the main concern in the EMT. There have been lots of models and methods employed for determining the effective constitutive parameters, either analytically or numerically [16–23]. For the cases of achiral inclusions, the well-known Maxwell-Garnet (MG) formula is widely used for obtaining the effective constitutive parameters in the long-wavelength limit due to its accuracy and efficiency [24]. Although the MG formula has already been extended to three-dimensional (3D) chiral metamaterials composed of isotropic spherical chiral particles fora longtime[25,26], the more nontrivial extension tobi-anisotropic cylinders in two-dimensional (2D) system (the system is translation invariant along out-of-plane direction) is still lacking. Unlike the 3D isotropic chiral metamaterials, the 2D bi-anisotropic metamaterials is inherently anisotropic due to the asymmetry between the in-plane and out-of-plane directions even if the inclusions are isotropic. This structural anisotropy as well as the material anisotropy of the inclusion itself bring us great difficulty to obtain a close-form formula when applying the EMT and conventionally the effective constitutive parameters are only obtainable by numerical retrieval [27–33].

In this work, we derived for the first time the analytic formulas of the effective constitutive parameters for 2D bi-anisotropic metamaterials whose chirality tensors have both diagonal and off-diagonal terms, in the long-wavelength limit through the use of Mie theory. Two sets of formulas were derived, one is based on the Mie coefficients of the inclusions which are valid even for inclusions having complex underlying structures, the other is based on the constitutive parameters of the homogeneous inclusions. The latter set of formulas are directly reduced to the traditional MG formula for 2D systems in the limit of vanishing chirality. Consequently, the formulas can be regarded as an extension of the MG formula to the 2D bi-anisotriopic metamaterials. The validities of our analytical formulas are confirmed by the full wave numerical simulations. Our work will benefit the investigation of wave manipulating properties of the bi-anisotropic metamaterials, and will also be conductive to the design of metamaterials for practical uses.

2. Helmhotz equation for the bi-anisotropic medium

The geometrical sketch of the bianisotropic metamaterial considered in this study are schematically shown in Fig. 1, where identical bi-anisotropic inclusions are arranged into a regular or random lattice in the xy plane (in-plane) embedded in air with permittivity ε0 and permeability µ0. The inclusions are invariant along z (out-of-plane) direction, and only the in-plane propagating electromagnetic waves (the out-of-plane wavenumber kz = 0) are considered throughout this paper. Therefore, the electromagnetic fields are independent of z.

 figure: Fig. 1

Fig. 1 Top view of the bianisotropic metamaterial with parallel cylindrical inclusions arranged in either (a) regular or (b) random lattice in the xy plane. The in-plane and out-of-plane directions are along xy and z directions, respectively.

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The constitutive relations of the bi-anisotropic medium are given by [34]

D=ε0ε¯¯⋅E+icκ¯¯⋅H,B=μ0μ¯¯⋅H−icκ¯¯T⋅E,
where c is the light speed in vacuum, and ε¯¯,μ¯¯, κ¯¯ are the relative permittivity, permeability and chirality tensors. For the sake of simplicity, in the following we will set ϵ0 = µ0 = c = 1. We assume that the medium is reciprocal (ε¯¯=ε¯¯T,μ¯¯=μ¯¯T) and isotropic in the xy plane. Therefore the permittivity and permeability tensors can be written as
ε¯¯=εtx^x^+εty^y^+εzz^z^,μ¯¯=μtx^x^+μty^y^+μzz^z^,
where εt,εz (µt,µz) are the in-plane and out-of-plane relative permittivities (permeabilities). The chirality tensor considered here has both diagonal and off-diagonal terms and is written as
κ¯¯=κtx^x^+κty^y^+κzz^z^+κkx^y^−κky^x^.

We will show that the electromagnetic wave equation inside this medium can be written into a Helmholtz form. Therefore, the fields can be expanded in terms of the vector cylindrical wave functions (VCWFs) [35].

According to the Maxwell’s equations and note that the system is invariant along z direction, namely, ∂z f = 0, where f is any field component, we can obtain that

∇⋅D=∇⋅(ε¯¯⋅E+iκ¯¯⋅H)=εt∇⋅E+iκt∇⋅H+iκk(∇×H)⋅z^=0,∇⋅B=∇⋅(μ¯¯⋅H−iκ¯¯T⋅H)=μt∇⋅H−iκt∇⋅E+iκk(∇×E)⋅z^=0,∇×D=∇×(ε¯¯⋅E+iκ¯¯T⋅H)=ε¯¯′⋅∇×E+iκ¯¯′⋅∇×H−iκk∇⋅Hz^,∇×B=∇×(μ¯¯⋅H−iκ¯¯T⋅H)=μ¯¯′⋅∇×H−iκ¯¯′⋅∇×E−iκk∇⋅Ez^,
where
ε¯¯′=εz(x^x^+y^y^)+εtz^z^,μ¯¯′=μz(x^x^+y^y^)+μtz^z^,κ¯¯′=κz(x^x^+y^y^)+κtz^z^.

Note that ∇×E = ik0B, ∇×H = −ik0D, where k0 = ω/c is the wavenumber in vacuum, and combine the first two subequations in Eq. (4), we obtain that

∇⋅E=−ik0κk(κtBz−iμtDz)εtμt−κt2,∇⋅H=−ik0κk(κtDz−iεtBz)εtμt−κt2.

We can see that the electromagnetic fields are divergent-less only when κk = 0. Substituting Eq. (6) into Eq. (4), the last two subequations become

∇×D=k0(M¯¯1⋅D+iM¯¯2⋅B),∇×B=k0(M¯¯1⋅B−iM¯¯3⋅D),
where
M¯¯1=κz(x^x^+y^y^)+εtμt−κt2−κk2εtμt−κt2κtz^z^,M¯¯2=εz(x^x^+y^y^)+εtμt−κt2−κk2εtμt−κt2εtz^z^,M¯¯3=μz(x^x^+y^y^)+εtμt−κt2−κk2εtμt−κt2μtz^z^.

Therefore,

∇×D=−∇2D=k02(M˜1⋅M¯¯1+M˜2⋅M¯¯3)⋅D+ik02(M˜1⋅M¯¯2+M˜2⋅M¯¯1)⋅B,∇×B=−∇2B=k02(M˜1⋅M¯¯1+M˜3⋅M¯¯2)⋅B−ik02(M˜1⋅M¯¯3+M˜3⋅M¯¯1)⋅D,
where we have used ∇×D = ∇ (∇·D) − ∇2D = −∇2D, and
M˜1=εtμt−κt2−κk2εtμt−κt2κt(x^x^+y^y^)+κzz^z^,M˜2=εtμt−κt2−κk2εtμt−κt2εt(x^x^+y^y^)+εzz^z^,M˜3=εtμt−κt2−κk2εtμt−κt2μt(x^x^+y^y^)+μzz^z^.

We can formulate Eq. (9) into a Helmholtz equation as

∇2(DB)+k02(M˜1⋅M¯¯1+M˜2⋅M¯¯3i(M˜1⋅M¯¯2+M˜2⋅M¯¯1)M˜1⋅M¯¯1+M˜3⋅M¯¯3i(M˜1⋅M¯¯3+M˜3⋅M¯¯1))(DB)=0,
whose basic solutions are the VCWFs.

3. Mie coefficients of the bi-anisotropic cylinder

3.1. κk = 0

We first consider that the chirality tensor has no off-diagonal terms, i.e., κk = 0. This kind of bi-anisotropic metamaterial can be realized using a lattice of metallic helical structures [1, 4] and were used for realizing Weyl metamaterials [13, 14]. In the absence of κk, the Helmholtz equation Eq. (11) can be greatly simplified. Note that the in-plane fields are decoupled from the out-of-plane fields, we only need to consider the wave equation for the out-of-plane fields. Using the constitutive relations in Eq. (1), we formulated the eigenvalue problem as

k2(EzHz)=k02(εzμt+κzκtiμtκz+iμzκt−iεtκz−iεzκtεtμz+κzκt)(EzHz).

The two wavenumbers inside the medium are determined by the eigenvalues of the 2 by 2 matrix in Eq. (12) which are

2(k±/k0)2=εzμt+εtμz+2κzκt±Δ,Δ=(εzμt−εtμz)2+4κzκt(εzμt+εtμz)+4κz2εtμt+4κt2εzμz,
and the corresponding polarization settings are determined by the corresponding eigenvectors
(EzHz)∝(i(εzμt−εtμz±Δ)2(κzμt+κtμz)1)=(iv±1).

Consider that an electromagnetic wave is scattered by a bi-anisotropic cylinder with radius r0. Based on the idea of Mie theory [35], we expanded the electric field inside the cylinder in series of the VCWFs as

Eins(r)=∑n[cnNn(1)(k+,r)+cn′Mn(1)(k+,r)+dnNn(1)(k−,r)+dn′Mn(1)(k−,r)],
where r = (r,ϕ) is the cylindrical coordinate with the origin located at the cylinder’s center, cn,cn′,dn,dn′ are expansion coefficients to be determined, and Ln(J),Mn(J),Nn(1) are the VCWFs with explicit expressions given by [35, 36]
Ln(J)=[inρzn(J)(ρ)ϕ^+zn(J)′(ρ)r^]einϕ,Mn(J)=[inρzn(J)(ρ)r^+zn(J)′(ρ)ϕ^]einϕ,Nn(J)=zn(J)(ρ)einϕz^,

In Eq. (16), ρ = kr, zn(1)=Jn and zn(3)=Hn(1) are the Bessel function and Hankel function of the first kind, and zn(J)′=dzn(J)(ρ)/dρ. The curls and divergences of VCWFs satisfy the following relations [35]

∇×Ln(J)=∇⋅Mn(J)=∇⋅Nn(J)=0,∇×Mn(J)=−∇⋅Ln(J)z^=kNn(J),∇×Nn(J)=kMn(J).

Note that the electromagnetic fields are divergent-less for κk = 0, according to Eq. (17), they are expanded only using Mn(J) and Nn(J), see Eq. (15). The magnetic field inside the cylinder can be obtained according to

∇×Eins=iωBins=iω(μ¯¯⋅Hins−iκ¯¯⋅Eins),
which gives
Hins=−iμt∑n[k+cn−κtcn′)Mn(1)(k+,r)+(k−dn−κtdn′)Mn(1)(k−,r)+(k+cn′−κzcn)Nn(1)(k+,r)+(k−dn′−κzdn)Nn(1)(k−,r)].

According to the polarization settings defined in Eq. (14), and combing the expressions of the electric (Eq. (15)) and magnetic (Eq. (19)) fields, we obtained that

iμzcn=iv+(k+cn′−κzcn)iμzdn=iv−(k−dn′−κzdn).

Assuming the background is air, the incident (Ei, Hi) and scattered (Es, Hs) fields can be expressed as

Ei=∑n[qnNn(1)(k0,r)+pnMn(1)(k0,r)],Hi=−i∑n[pnNn(1)(k0,r)+qnMn(1)(k0,r)],Es=−∑n[bnNn(3)(k0,r)+anMn(3)(k0,r)],Hs=i∑n[anNn(3)(k0,r)+bnMn(3)(k0,r)],
where pn, qn are beam shape coefficients and an, bn are scattering coefficients. They are related by the Mie coefficients as [35]
an=Anpn+Cnqn,bn=Dnpn+Bnqn.

Applying the boundary conditions that the tangential electromagnetic field components should be continuous at r = r0,

qnJn(x0)−bnHn(1)(x0)=cnJn(x+)+dnJn(x),pnJn′(x0)−anHn(1)′(x0)=cn′Jn′(x+)−dn′Jn′(x−),pnJn(x0)−anHn(1)(x0)=μz−1[(k+cn′−κzcn)Jn(x+)+(k−dn′−κzdn)Jn(x−)],qnJn′(x0)−bnHn(1)′(x0)=μt−1[(k+cn−κtcn′)Jn′(x+)+(k−dn−κtdn′)Jn′(x−)],
where x0 = k0r0, x± = k±r0 are the dimensionless size parameters, the Mie coefficients for the bi-anisotropic cylinder are obtained by solving Eqs. (20), (22) and (23). In the long-wavelength limit where x0 ≪ 1, the zeroth and first order Mie coefficients are
A0=−i4(μz−1)πx02,A±1=i4(1−εt)(1+μt)+κt2(1+εt)(1+μt)−κt2πx02,B0=−i4(εz−1)πx02,B±1=i4(1+εt)(1−μt)+κt2(1+εt)(1+μt)−κt2πx02,C0=D0=−i4κzπx02,C±=D±1=−i2κt(1+εt)(1+μt)−κt2πx02.

3.2. κt = κz = 0

Next we consider that the chirality tensor possesses no diagonal terms, i.e., κt = κz = 0. This kind of bi-anisotropic metamaterial canbe realized using alattice of split-ring resonators [2, 10] or Ω-particles [37–39]. The matrix in Eq. (11) becomes diagonal. It is easy to obtain the two wavenumbers inside the medium as

(k1/k0)2=εzεt−1(εtμt−κk2),(k2/k0)2=μzμt−1(εtμt−κk2),
and the corresponding polarization settings are Dx = Dy = Bz = 0 and Dz = Bx = By = 0, respectively. Therefore, the electric displacement and magnetic induced fields can be expanded as
Dins=∑n[dnNn(1)(k1,r)+enMn(1)(k2,r),Bins=∑n[cnMn(1)(k1,r)+fnNn(1)(k2,r),
where cn, dn, en, fn are expansion coefficients, According to Eq. (7) and using the properties of VCWFs, see Eq. (17), we obtained that
dnk1=iεzcn,enk2=i(εt−κk2μt)fn,
and the electromagnetic fields inside the medium as
Eins=∑ndnεzNn(1)(k1,r)+μtenεtμt−κk2Mn(1)(k2,r)+iκkcnεtμt−κk2Ln(1)(k1,r),Hins=∑nfnμzNn(1)(k2,r)+εtenεtμt−κk2Mn(1)(k1,r)+iκkenεtμt−κk2Ln(1)(k2,r).

The electromagnetic fields are non longer divergent-less because of the nonzero κk. Also imposing the boundary conditions,

qnJn(x0)−bnHn(1)(x0)=εz−1dnJn(x1),−ipnJn(x0)+ianHn(1)(x0)=μz−1fnJn(x2),pnJn′(x0)−anHn(1)′(x0)=μtenεtμt−κk2Jn′(x2)+κkcnεtμt−κk2nx1Jn(x1),−iqnJn′(x0)+ibnHn(1)′(x0)=εtcnεtμt−κk2Jn′(x1)+κkenεtμt−κk2nx2Jn(x2),
where x1,2 = k1,2r0, the Mie coefficients are obtained by solving Eqs. (22), (27) and (29). In the long-wavelength limit, the zeroth and first order Mie coefficients are
A0=−i4(μz−1)πx02,A±1=i4(1−εt)(1+μt)+κk2(εt+1)(μt+1)−κk2πx02,B0=−i4(εz−1)πx02,B±1=i4(1+εt)(1−μt)+κk2(εt+1)(μt+1)−κk2πx02,C0=D0=0,C±1=D±1=±12κk(εt+1)(μt+1)−κk2πx02.

Combining Eqs. (24) and (30), we can conclude that the general expressions of Mie coefficients for bi-anisotropic cylinders with chirality tensor possessing both diagonal and off-diagonal terms are

A0=−i4(μz−1)πx02,A±1=i4(1−εt)(1+μt)+κk2(εt+1)(μt+1)−κk2πx02,B0=−i4(εz−1)πx02,B±1=i4(1+εt)(1−μt)+κk2(εt+1)(μt+1)−κk2πx02,C0=D0=−i4κzπx02,C±1=D±1=12−iκt±κk(εt+1)(μt+1)−κk2πx02.

4. Closed-form experssions for the effective constitutive parameters

Consider a metamaterial composed of identical bi-anisotropic inclusions arranged into a regular or random lattice embedded in air. The lattice constant fulfills k0a ≪ 1. Therefore, the local effective medium theory, such as the coherent potential approximation (CPA) method, can be applied due to the weak multiple scattering. Following the idea of CPA method [17], the zeroth and first order Mie coefficients of a single bi-anisotropic inclusion should equal to those of an effective cylinder with the effective constitutive parameters and occupying a unit cell volume, namely

Aie=Ai,Bie=Bi,Cie=Ci,Die=Di,i=−1,0,1
where Ai, Bi, Ci, Di are Mie coefficients of the inclusion, and Aie,Bie,Cie,Die are the Mie coefficients of the effective cylinder calculated by substituting the effective constitutive parameters into Eq. (31). According to Eqs. (31) and (32), we can obtain the effective constitutive parameters as
εez=1+iΛB0,μez=1+iΛA0,κez=iΛC0,εet=(i−ΛA1)(i+ΛB1)+Λ2C1D1(i+ΛA1)(i+ΛB1)−Λ2C1D1,μet=(i+ΛA1)(i−ΛB1)+Λ2C1D1(i+ΛA1)(i+ΛB1)−Λ2C1D1,κet=iΛ2(C1−D1)(i+ΛA1)(i+ΛB1)−Λ2C1D1,κek=Λ(C1−D1)(i+ΛA1)(i+ΛB1)−Λ2C1D1,
where Λ=4/k02Ω with Ω being the volume of a unit cell. If the inclusion is cylindrical and homogeneous, substituting Eq. (31) into Eq. (33), we obtain that
εez=(εz−1)p+1,μez=(μz−1)p+1,κez=κzp,εet=(1+εt)(1+μt)−κt2−κk2+2(εt−μt)p−[(1−εt)(1−μt)−κt2−κk2]p2(1+εt)(1+μt)−κt2−κk2+2(1−εtμt+κt2+κt2)p+[(1−εt)(1−μt)−κt2−κk2]p2,μet=(1+εt)(1+μt)−κt2−κk2+2(εt−μt)p−[(1−εt)(1−μt)−κt2−κk2]p2(1+εt)(1+μt)−κt2−κk2+2(1−εtμt+κt2+κt2)p+[(1−εt)(1−μt)−κt2−κk2]p2,κet=4κtp(1+εt)(1+μt)−κt2−κk2+2(1−εtμt+κt2+κt2)p+[(1−εt)(1−μt)−κt2−κk2]p2κek=4κkp(1+εt)(1+μt)−κt2−κk2+2(1−εtμt+κt2+κt2)p+[(1−εt)(1−μt)−κt2−κk2]p2,
where p=πx02/k02Ω is the filling ratio. For κk = 0, and consider the inclusion to be isotropic (its permittivity, permeability and chirality are scalars), Eqs. (33) and (34) are just reduced to Eqs. (20) and (22) in our previous work [40], which were derived using the multiple scattering technique. Eq. (34) is reduced to the well-known MG formula in the absence of chirality (κt = κz = κk = 0). Therefore, we can regard Eq. (34) as the extended MG formula for bi-anistropic metamateials. We note that although Eq. (33) is less efficient than Eq. (34) in calculating the effective constitutive parameters, it is more fundamental since it does not require the inclusion to be homogeneous. For the inclusion whose Mie coefficients satisfy A1 = A−1, B1 = B−1, C±1 = D∓1, Eq. (33) can be applied, even if the inclusion has complex underlying structures, e.g., the inclusion is made of helices [40]. The Mie coefficients of inclusion having complex structures need to be retrieved using the scattering fields in the far field region. However, the retrieval of the Mie coefficients only needs to be done once if the metamaterials with different filling ratios and lattice symmetries are composed of the same inclusions. This is a great advantage comparing with conventional parameter retrieval methods, which requires the repetition of the retrieval procedures when the filling ratio or lattice symmetry changes.

Note that Eqs. (33) and (34) are valid only when the long-wavelength approximation holds, which requires that the incident wavelength is about 10 times larger than the lattice constant of the metamaterial. Therefore, the frequency usually should not be greater than c/(10a). When the frequency is too high, the effective constitutive parameters can only be obtained by the numerical retrieval, because for this case no analytic expressions can be derived due to the non-local effect and boundary sensitivity. Eqs. (33) and (34) are valid irrespective of the material parameters of the inclusions and the lattice structures. When the filling ratio is not too large, i.e., p < 0.5, Eqs. (33) and (34) are always accurate. However, if the filling ratio is too high, i.e., p > 0.5, the multiple scattering effect between cylinders will become important. Then the multiple scattering technique should be used and the high filling ratio corrections to the formulas should be taken into account [41].

5. Numerical testing

In this section, we will test the validity of our formula Eq. (34) using the full wave numerical simulations. Consider a plane wave obliquely incident on a metamaterial slab which is infinite along y direction and has a finite width d = 50a along x direction, where a is the lattice constant of the square lattice that the bi-anisotropic cylindrical inclusions are arranged into, see the inset in Fig. 2(a). For comparison, we also considered the same plane wave illuminates on an effective medium slab possessing the same width d and effective constitutive parameters calculated by Eq. (34). The validity of the effective parameters can be tested by checking the consistence between the spatially averaged lattice fields, such as the electric field or displacement field: E¯=∫ΩEdxdy/Ω, D¯=∫ΩDdxdy/Ω, where the spatial integrals are over the whole unit cell, and the fields in the effective medium Ee, De, since these two are equal in the long-wavelength limit [42,43].

In Fig. 2, we showed the spatially averaged lattice fields in the metamaterials having the square lattice structure and the corresponding fields in effective medium by circles and lines, respectively. All the fields are calculated using the finite-element method package COMSOL [44]. We can see that for different incident angles, the spatially averaged lattice fields (circles) both inside and outside the slab are in accordance with the fields in the effective medium (lines). Such consistence also exists for other field components and polarized incident waves. This indicates that our formula Eq. (34) can indeed determine the effective constitutive parameters correctly for this kind of bianisotropic metamaterials.

 figure: Fig. 2

Fig. 2 The spatially averaged electric field and displacement field of each layer inside the metamaterials having the square lattice structure for the incident angle (a) θ = π/6 and (b) θ = π/3 are noted by circles, while the electric field and displacement field along the x direction in corresponding effective continuous medium under the same incidence are shown by lines. The effective medium slab (photonic crystal) region is between the dotted lines. The incident beam is Ez polarized with amplitude E0. The constitutive parameters of the cylinder are εt = 2,εz = 5,µt = µz = 1,κt = 0.5,κz = 0.3,κk = 0.4, and the radius is r0 = 0.3a.

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In Fig. 3, we also compared the spatially averaged lattice fields in the metamaterials having a triangular lattice structure and the corresponding fields in effective medium slab. Now 60 layers (d=303a) are used. The excellent agreement is also seen. In fact, because our formulas is obtained based on the local EMT, they are valid irrespective of the lattice symmetry.

 figure: Fig. 3

Fig. 3 The spatially averaged electric field and displacement field of each layer inside the metamaterials having the triangular lattice structure for the incident angle (a) θ = π/6 and (b) θ = π/3 are noted by circles, while the electric field and displacement field along the x direction in corresponding effective continuous medium under the same incidence are shown by lines. The effective medium slab (photonic crystal) region is between the dotted lines. The incident beam is Ez polarized with amplitude E0. The constitutive parameters of the cylinder are εt = 8,εz = 3, µt = µz = 1, κt = 0.3, κz = 0.4, κk = 0.5, and the radius is r0 = 0.3a.

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The validity of Eq. (34) can also be tested according to the effective refractive indices, which are calculated either according to the slopes of the band dispersion of the metamaterial possessing a lattice structure, i.e., n/c = dk/dω in the limit of ω →0, k →0, or by n = k/k0 which are obtained by substituting the effective constitutive parameters into Eq. (13) or (25). The results by these two methods should be equal if the effective constitutive parameters are correct.

The refractive indices calculated by these two methods for κk = 0 and κt = κz = 0 are shown in Figs. 4(a) and 4(b), respectively. We first used Eq. (34) to calculate the effective constitutive parameters and then substituted them into Eqs. (13) and (25) to obtain the refractive indices analytically. The band dispersion for the square lattice is obtained using COMSOL. It is seen that the numerical results based on the band dispersion analysis (circles) match the analytic results using the formulas very well, again confirming the validity of Eq. (34).

 figure: Fig. 4

Fig. 4 Comparison between the refractive indices obtained from the formulas (lines) and those from the slopes of band dispersion (circles). (a) The chirality tensor of the cylinder has no off-diagonal terms (κk = 0) and κz = 0.5. (b) The chirality tensor of the cylinder has no diagonal terms (κt = κz = 0). The other constitutive parameters of the cylinder are εt = 10, εz = 5, µt = µz = 1 and the radius is r0 = 0.35a.

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The cases of strong resonances were also investigated. In Fig. 5, we showed the refractive indices calculated by these two methods when the cylinder is around the surface plasmon resonance. Resonances with very high quality factors occurr at κt ≈ 0.9 and κk ≈ 0.7 for Figs. 5(a) and 5(b), respectively. We can see that even very close to the resonance peak where the refractive index goes to infinity, the formulas can still produce the effective refractive indices accurately, indicating that our formulas are also valid for strong resonance cases. Note that the unplotted ranges for n− (κt < 0.9) and n2(κk < 0.7) are corresponding to the photonic band gaps for n− and n2, respectively.

 figure: Fig. 5

Fig. 5 Comparison between the refractive indices obtained from the formulas (lines) and those from the slopes of band dispersion (circles). (a) The chirality tensor of the cylinder has no off-diagonal and out-of-plane terms (κk = κz = 0). (b) The chirality tensor of the cylinder has no diagonal terms (κt = κz = 0). The other constitutive parameters of the cylinder are εt = εz = −1.5, µt = µz = 1 and the radius is r0 = 0.3a.

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6. Conclusion

In summary, by deriving the expressions of Mie coefficients and using the CPA method, we derived the closed-form expressions for the effective constitutive parameters of 2D bi-anisotropic metamaterials in the long-wavelength limit. Our formulas work for reciprocal metamaterials whose chirality tensors can have both diagonal and off-diagonal terms and satisfy κxy = −κyx = κk. The formulas can be regarded as the extended MG formulas for bi-anisotropic metamaterials since they are directly reduced to the traditional MG formulas when the chirality is vanishing. In principle, our method for deriving the closed-form formulas can be extended to the case that metamaterials are composed of non-reciprocal inclusions, e.g., the permittivity tensor possesses pure imaginary off-diagonal components, although the Mie coefficients will become more complicated. However, for κxy = κyx, our method fails because the wave equation inside the medium is no longer the Helmholtz form and the fields cannot be expanded in series of VCWFs.

Funding

Natural Science Foundation of SZU (2019009); National Natural Science Foundation of China (NSFC) (11734012 and 11574218).

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Figures (5)
Fig. 1
Fig. 1 Top view of the bianisotropic metamaterial with parallel cylindrical inclusions arranged in either (a) regular or (b) random lattice in the xy plane. The in-plane and out-of-plane directions are along xy and z directions, respectively.

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Fig. 2
Fig. 2 The spatially averaged electric field and displacement field of each layer inside the metamaterials having the square lattice structure for the incident angle (a) θ = π/6 and (b) θ = π/3 are noted by circles, while the electric field and displacement field along the x direction in corresponding effective continuous medium under the same incidence are shown by lines. The effective medium slab (photonic crystal) region is between the dotted lines. The incident beam is Ez polarized with amplitude E0. The constitutive parameters of the cylinder are εt = 2,εz = 5,µt = µz = 1,κt = 0.5,κz = 0.3,κk = 0.4, and the radius is r0 = 0.3a.

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Fig. 3
Fig. 3 The spatially averaged electric field and displacement field of each layer inside the metamaterials having the triangular lattice structure for the incident angle (a) θ = π/6 and (b) θ = π/3 are noted by circles, while the electric field and displacement field along the x direction in corresponding effective continuous medium under the same incidence are shown by lines. The effective medium slab (photonic crystal) region is between the dotted lines. The incident beam is Ez polarized with amplitude E0. The constitutive parameters of the cylinder are εt = 8,εz = 3, µt = µz = 1, κt = 0.3, κz = 0.4, κk = 0.5, and the radius is r0 = 0.3a.

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Fig. 4
Fig. 4 Comparison between the refractive indices obtained from the formulas (lines) and those from the slopes of band dispersion (circles). (a) The chirality tensor of the cylinder has no off-diagonal terms (κk = 0) and κz = 0.5. (b) The chirality tensor of the cylinder has no diagonal terms (κt = κz = 0). The other constitutive parameters of the cylinder are εt = 10, εz = 5, µt = µz = 1 and the radius is r0 = 0.35a.

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Fig. 5
Fig. 5 Comparison between the refractive indices obtained from the formulas (lines) and those from the slopes of band dispersion (circles). (a) The chirality tensor of the cylinder has no off-diagonal and out-of-plane terms (κk = κz = 0). (b) The chirality tensor of the cylinder has no diagonal terms (κt = κz = 0). The other constitutive parameters of the cylinder are εt = εz = −1.5, µt = µz = 1 and the radius is r0 = 0.3a.

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Equations (34)

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D = ε 0 ε ¯ ¯ ⋅ E + i c κ ¯ ¯ ⋅ H , B = μ 0 μ ¯ ¯ ⋅ H − i c κ ¯ ¯ T ⋅ E ,
ε ¯ ¯ = ε t x ^ x ^ + ε t y ^ y ^ + ε z z ^ z ^ , μ ¯ ¯ = μ t x ^ x ^ + μ t y ^ y ^ + μ z z ^ z ^ ,
κ ¯ ¯ = κ t x ^ x ^ + κ t y ^ y ^ + κ z z ^ z ^ + κ k x ^ y ^ − κ k y ^ x ^ .
∇ ⋅ D = ∇ ⋅ ( ε ¯ ¯ ⋅ E + i κ ¯ ¯ ⋅ H ) = ε t ∇ ⋅ E + i κ t ∇ ⋅ H + i κ k ( ∇ × H ) ⋅ z ^ = 0 , ∇ ⋅ B = ∇ ⋅ ( μ ¯ ¯ ⋅ H − i κ ¯ ¯ T ⋅ H ) = μ t ∇ ⋅ H − i κ t ∇ ⋅ E + i κ k ( ∇ × E ) ⋅ z ^ = 0 , ∇ × D = ∇ × ( ε ¯ ¯ ⋅ E + i κ ¯ ¯ T ⋅ H ) = ε ¯ ¯ ′ ⋅ ∇ × E + i κ ¯ ¯ ′ ⋅ ∇ × H − i κ k ∇ ⋅ H z ^ , ∇ × B = ∇ × ( μ ¯ ¯ ⋅ H − i κ ¯ ¯ T ⋅ H ) = μ ¯ ¯ ′ ⋅ ∇ × H − i κ ¯ ¯ ′ ⋅ ∇ × E − i κ k ∇ ⋅ E z ^ ,
ε ¯ ¯ ′ = ε z ( x ^ x ^ + y ^ y ^ ) + ε t z ^ z ^ , μ ¯ ¯ ′ = μ z ( x ^ x ^ + y ^ y ^ ) + μ t z ^ z ^ , κ ¯ ¯ ′ = κ z ( x ^ x ^ + y ^ y ^ ) + κ t z ^ z ^ .
∇ ⋅ E = − i k 0 κ k ( κ t B z − i μ t D z ) ε t μ t − κ t 2 , ∇ ⋅ H = − i k 0 κ k ( κ t D z − i ε t B z ) ε t μ t − κ t 2 .
∇ × D = k 0 ( M ¯ ¯ 1 ⋅ D + i M ¯ ¯ 2 ⋅ B ) , ∇ × B = k 0 ( M ¯ ¯ 1 ⋅ B − i M ¯ ¯ 3 ⋅ D ) ,
M ¯ ¯ 1 = κ z ( x ^ x ^ + y ^ y ^ ) + ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 κ t z ^ z ^ , M ¯ ¯ 2 = ε z ( x ^ x ^ + y ^ y ^ ) + ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 ε t z ^ z ^ , M ¯ ¯ 3 = μ z ( x ^ x ^ + y ^ y ^ ) + ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 μ t z ^ z ^ .
∇ × D = − ∇ 2 D = k 0 2 ( M ˜ 1 ⋅ M ¯ ¯ 1 + M ˜ 2 ⋅ M ¯ ¯ 3 ) ⋅ D + i k 0 2 ( M ˜ 1 ⋅ M ¯ ¯ 2 + M ˜ 2 ⋅ M ¯ ¯ 1 ) ⋅ B , ∇ × B = − ∇ 2 B = k 0 2 ( M ˜ 1 ⋅ M ¯ ¯ 1 + M ˜ 3 ⋅ M ¯ ¯ 2 ) ⋅ B − i k 0 2 ( M ˜ 1 ⋅ M ¯ ¯ 3 + M ˜ 3 ⋅ M ¯ ¯ 1 ) ⋅ D ,
M ˜ 1 = ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 κ t ( x ^ x ^ + y ^ y ^ ) + κ z z ^ z ^ , M ˜ 2 = ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 ε t ( x ^ x ^ + y ^ y ^ ) + ε z z ^ z ^ , M ˜ 3 = ε t μ t − κ t 2 − κ k 2 ε t μ t − κ t 2 μ t ( x ^ x ^ + y ^ y ^ ) + μ z z ^ z ^ .
∇ 2 ( D B ) + k 0 2 ( M ˜ 1 ⋅ M ¯ ¯ 1 + M ˜ 2 ⋅ M ¯ ¯ 3 i ( M ˜ 1 ⋅ M ¯ ¯ 2 + M ˜ 2 ⋅ M ¯ ¯ 1 ) M ˜ 1 ⋅ M ¯ ¯ 1 + M ˜ 3 ⋅ M ¯ ¯ 3 i ( M ˜ 1 ⋅ M ¯ ¯ 3 + M ˜ 3 ⋅ M ¯ ¯ 1 ) ) ( D B ) = 0 ,
k 2 ( E z H z ) = k 0 2 ( ε z μ t + κ z κ t i μ t κ z + i μ z κ t − i ε t κ z − i ε z κ t ε t μ z + κ z κ t ) ( E z H z ) .
2 ( k ± / k 0 ) 2 = ε z μ t + ε t μ z + 2 κ z κ t ± Δ , Δ = ( ε z μ t − ε t μ z ) 2 + 4 κ z κ t ( ε z μ t + ε t μ z ) + 4 κ z 2 ε t μ t + 4 κ t 2 ε z μ z ,
( E z H z ) ∝ ( i ( ε z μ t − ε t μ z ± Δ ) 2 ( κ z μ t + κ t μ z ) 1 ) = ( i v ± 1 ) .
E i n s ( r ) = ∑ n [ c n N n ( 1 ) ( k + , r ) + c n ′ M n ( 1 ) ( k + , r ) + d n N n ( 1 ) ( k − , r ) + d n ′ M n ( 1 ) ( k − , r ) ] ,
L n ( J ) = [ i n ρ z n ( J ) ( ρ ) ϕ ^ + z n ( J ) ′ ( ρ ) r ^ ] e i n ϕ , M n ( J ) = [ i n ρ z n ( J ) ( ρ ) r ^ + z n ( J ) ′ ( ρ ) ϕ ^ ] e i n ϕ , N n ( J ) = z n ( J ) ( ρ ) e i n ϕ z ^ ,
∇ × L n ( J ) = ∇ ⋅ M n ( J ) = ∇ ⋅ N n ( J ) = 0 , ∇ × M n ( J ) = − ∇ ⋅ L n ( J ) z ^ = k N n ( J ) , ∇ × N n ( J ) = k M n ( J ) .
∇ × E i n s = i ω B i n s = i ω ( μ ¯ ¯ ⋅ H i n s − i κ ¯ ¯ ⋅ E i n s ) ,
H i n s = − i μ t ∑ n [ k + c n − κ t c n ′ ) M n ( 1 ) ( k + , r ) + ( k − d n − κ t d n ′ ) M n ( 1 ) ( k − , r ) + ( k + c n ′ − κ z c n ) N n ( 1 ) ( k + , r ) + ( k − d n ′ − κ z d n ) N n ( 1 ) ( k − , r ) ] .
i μ z c n = i v + ( k + c n ′ − κ z c n ) i μ z d n = i v − ( k − d n ′ − κ z d n ) .
E i = ∑ n [ q n N n ( 1 ) ( k 0 , r ) + p n M n ( 1 ) ( k 0 , r ) ] , H i = − i ∑ n [ p n N n ( 1 ) ( k 0 , r ) + q n M n ( 1 ) ( k 0 , r ) ] , E s = − ∑ n [ b n N n ( 3 ) ( k 0 , r ) + a n M n ( 3 ) ( k 0 , r ) ] , H s = i ∑ n [ a n N n ( 3 ) ( k 0 , r ) + b n M n ( 3 ) ( k 0 , r ) ] ,
a n = A n p n + C n q n , b n = D n p n + B n q n .
q n J n ( x 0 ) − b n H n ( 1 ) ( x 0 ) = c n J n ( x + ) + d n J n ( x ) , p n J n ′ ( x 0 ) − a n H n ( 1 ) ′ ( x 0 ) = c n ′ J n ′ ( x + ) − d n ′ J n ′ ( x − ) , p n J n ( x 0 ) − a n H n ( 1 ) ( x 0 ) = μ z − 1 [ ( k + c n ′ − κ z c n ) J n ( x + ) + ( k − d n ′ − κ z d n ) J n ( x − ) ] , q n J n ′ ( x 0 ) − b n H n ( 1 ) ′ ( x 0 ) = μ t − 1 [ ( k + c n − κ t c n ′ ) J n ′ ( x + ) + ( k − d n − κ t d n ′ ) J n ′ ( x − ) ] ,
A 0 = − i 4 ( μ z − 1 ) π x 0 2 , A ± 1 = i 4 ( 1 − ε t ) ( 1 + μ t ) + κ t 2 ( 1 + ε t ) ( 1 + μ t ) − κ t 2 π x 0 2 , B 0 = − i 4 ( ε z − 1 ) π x 0 2 , B ± 1 = i 4 ( 1 + ε t ) ( 1 − μ t ) + κ t 2 ( 1 + ε t ) ( 1 + μ t ) − κ t 2 π x 0 2 , C 0 = D 0 = − i 4 κ z π x 0 2 , C ± = D ± 1 = − i 2 κ t ( 1 + ε t ) ( 1 + μ t ) − κ t 2 π x 0 2 .
( k 1 / k 0 ) 2 = ε z ε t − 1 ( ε t μ t − κ k 2 ) , ( k 2 / k 0 ) 2 = μ z μ t − 1 ( ε t μ t − κ k 2 ) ,
D i n s = ∑ n [ d n N n ( 1 ) ( k 1 , r ) + e n M n ( 1 ) ( k 2 , r ) , B i n s = ∑ n [ c n M n ( 1 ) ( k 1 , r ) + f n N n ( 1 ) ( k 2 , r ) ,
d n k 1 = i ε z c n , e n k 2 = i ( ε t − κ k 2 μ t ) f n ,
E i n s = ∑ n d n ε z N n ( 1 ) ( k 1 , r ) + μ t e n ε t μ t − κ k 2 M n ( 1 ) ( k 2 , r ) + i κ k c n ε t μ t − κ k 2 L n ( 1 ) ( k 1 , r ) , H i n s = ∑ n f n μ z N n ( 1 ) ( k 2 , r ) + ε t e n ε t μ t − κ k 2 M n ( 1 ) ( k 1 , r ) + i κ k e n ε t μ t − κ k 2 L n ( 1 ) ( k 2 , r ) .
q n J n ( x 0 ) − b n H n ( 1 ) ( x 0 ) = ε z − 1 d n J n ( x 1 ) , − i p n J n ( x 0 ) + i a n H n ( 1 ) ( x 0 ) = μ z − 1 f n J n ( x 2 ) , p n J n ′ ( x 0 ) − a n H n ( 1 ) ′ ( x 0 ) = μ t e n ε t μ t − κ k 2 J n ′ ( x 2 ) + κ k c n ε t μ t − κ k 2 n x 1 J n ( x 1 ) , − i q n J n ′ ( x 0 ) + i b n H n ( 1 ) ′ ( x 0 ) = ε t c n ε t μ t − κ k 2 J n ′ ( x 1 ) + κ k e n ε t μ t − κ k 2 n x 2 J n ( x 2 ) ,
A 0 = − i 4 ( μ z − 1 ) π x 0 2 , A ± 1 = i 4 ( 1 − ε t ) ( 1 + μ t ) + κ k 2 ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 , B 0 = − i 4 ( ε z − 1 ) π x 0 2 , B ± 1 = i 4 ( 1 + ε t ) ( 1 − μ t ) + κ k 2 ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 , C 0 = D 0 = 0 , C ± 1 = D ± 1 = ± 1 2 κ k ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 .
A 0 = − i 4 ( μ z − 1 ) π x 0 2 , A ± 1 = i 4 ( 1 − ε t ) ( 1 + μ t ) + κ k 2 ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 , B 0 = − i 4 ( ε z − 1 ) π x 0 2 , B ± 1 = i 4 ( 1 + ε t ) ( 1 − μ t ) + κ k 2 ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 , C 0 = D 0 = − i 4 κ z π x 0 2 , C ± 1 = D ± 1 = 1 2 − i κ t ± κ k ( ε t + 1 ) ( μ t + 1 ) − κ k 2 π x 0 2 .
A i e = A i , B i e = B i , C i e = C i , D i e = D i , i = − 1 , 0 , 1
ε e z = 1 + i Λ B 0 , μ e z = 1 + i Λ A 0 , κ e z = i Λ C 0 , ε e t = ( i − Λ A 1 ) ( i + Λ B 1 ) + Λ 2 C 1 D 1 ( i + Λ A 1 ) ( i + Λ B 1 ) − Λ 2 C 1 D 1 , μ e t = ( i + Λ A 1 ) ( i − Λ B 1 ) + Λ 2 C 1 D 1 ( i + Λ A 1 ) ( i + Λ B 1 ) − Λ 2 C 1 D 1 , κ e t = i Λ 2 ( C 1 − D 1 ) ( i + Λ A 1 ) ( i + Λ B 1 ) − Λ 2 C 1 D 1 , κ e k = Λ ( C 1 − D 1 ) ( i + Λ A 1 ) ( i + Λ B 1 ) − Λ 2 C 1 D 1 ,
ε e z = ( ε z − 1 ) p + 1 , μ e z = ( μ z − 1 ) p + 1 , κ e z = κ z p , ε e t = ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( ε t − μ t ) p − [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( 1 − ε t μ t + κ t 2 + κ t 2 ) p + [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 , μ e t = ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( ε t − μ t ) p − [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( 1 − ε t μ t + κ t 2 + κ t 2 ) p + [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 , κ e t = 4 κ t p ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( 1 − ε t μ t + κ t 2 + κ t 2 ) p + [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 κ e k = 4 κ k p ( 1 + ε t ) ( 1 + μ t ) − κ t 2 − κ k 2 + 2 ( 1 − ε t μ t + κ t 2 + κ t 2 ) p + [ ( 1 − ε t ) ( 1 − μ t ) − κ t 2 − κ k 2 ] p 2 ,
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