Role of photonic angular momentum states in nonreciprocal diffraction from magneto-optical cylinder arrays

In recent years, there has been an increased interest to magneto-optical (MO) effects in artificial optical nanostructures due to their exotic physical behaviors and potential applications.^1–15 The existence of nonzero off-diagonal tensor elements in either the magnetic permeability or the dielectric permittivity^9–15 breaks the time-reversal symmetry of an optical wave propagating inside a MO medium,¹¹ which is a necessary condition to design one-way waveguides, optical isolators and other nonreciprocal optical devices.^7,9–12,15 It is also shown that MO effects can manipulate the characteristics of extraordinary optical transmission through metamaterials, not only change the frequencies of resonant transmission peaks,^1,3–5,13 but also enhance the MO Faraday and Kerr effects.^2,6

It is well known that MO effects render different propagation constants or refractive indexes to the optical waves with opposite circular polarizations, an phenomenon that can be termed the circular birefringence.¹⁶ Nonreciprocal optical effects can be obtained by using circular birefringence to achieve different reflection or transmission coefficients of the left- and right-handed circular polarizations.¹⁶ By noting that optical polarization carries the spin angular momentum (SAM) of light,^16,17 it is then natural to question whether a similar effect can be realized on the other degree of freedoms in photons, the orbital angular momentum (OAM)¹⁷ associated with a helical phase wavefront. Artificially optical nanostructures provide a platform to realize such a possibility, where the eigenfrequencies of photonic angular momentum states (PAMSs)^14,15,18 featured by helical phases of exp (−jmϕ), are shown to be sensitive to the off-diagonal tensor elements. Here ϕ is the azimuthal angle in the cylindrical coordinate frames of (r, ϕ, z), and m is the quantized topological charge. Wang et. al. showed that a classic analogue of the Zeeman effect on PAMSs can be obtained in a MO cylinder,¹⁸ that the frequency degeneracy of ±m PAMSs is broken and the shift in frequency is determined by the topological charge m. Some useful mode-decoupling properties for photonics application¹⁸ have been demonstrated. Nonreciprocal optical diffraction effects, especially a nonreciprocal negative directional transmission associated with m = ±1 PAMSs in subwavelength MO cylinder arrays,¹⁵ have been demonstrated and briefly discussed.

In this article, we provide a comprehensive study on the influence of PAMSs, which are subjected to the classic analogue of the Zeeman effect, to the optical diffraction from MO cylinder arrays. We pay attention to the situation that high-m PAMSs are excited, and that the cylinder array is no long subwavelength and supports high diffraction orders even in normal incidence. We show that transmission of the 0th diffraction order is reciprocal, and nonreciprocal diffraction effects can be obtained only for these nonzero orders. Some exotic phenomena, such as a dispersionless quasi-omnidirectional nonreciprocal diffraction and transmission spikes associated with PAMS of a large m, are present and discussed. Role of PAMSs, the excitation of which are sensitive to the directions of incidence, applied magnetic field, and orientation of the cylinder arrays, are studied. This investigation provides deeper insights into the importance and various applications of PAMSs in different disciplines, which would contribute to the future demands in designing compact optical components for on-chip applications.

Before presenting the numerical simulation and analysis results, we would like to propose a theory to qualitatively explain why the diffraction from a MO cylinder array should be nonreciprocal. Figure 1 shows the required definitions of coordinate frames and geometric parameters of the MO cylinder arrays.

Consider the two situations shown in Fig. 1. When the angle of incidence θ is a positive one of +θ₀, see Fig. 1(a), the phase difference ΔΘ between two adjacent cylinders reads ΔΘ = k₀dsin θ₀, where k₀ = 2π/λ is the wavevector and λ is the free-space wavelength. To compensate for this phase difference, a proper rotating energy flux S such as the clockwise one shown in Fig. 1(a) should be excited in each cylinder. For the reciprocal case, where the angle of incidence is a negative one of −θ₀ [see Fig. 1(b)], the phase difference becomes to ΔΘ = −k₀dsin θ₀, and consequently a reversed rotating energy flux S is required.

Transverse rotating energy fluxes are associated with PAMSs of nonzero topological charges m, and the direction of rotation is correlated to the sign of m.^14,15,17 If PAMSs are degenerated, that for opposite m values the field distributions Ψ_m(r), resonant frequencies ω_m/2π and widthes Γ_m are identical, the two cases shown in Fig. 1 are reciprocal. To see it, let us assume that the excited field in each single MO cylinder of the array for an angle of incidence θ is

where Ψ_m(r) is the eigenfunction of the m PAMS representing its r-dependent distribution of field, and

$Cmθ$ is an expansion coefficient to be determined. When the material of the cylinder is optically isotropic, the ±m PAMSs are degenerated so that Ψ_−m(r) = Ψ_m(r). Additional, with the degenerated resonant conditions of ±m PAMSs in their eigenfrequencies ω_±m/2π and widthes Γ_±m, it is readily to get that

$C_{+m}^{+\theta _0}=C_{-m}^{-\theta _0}$

$C+m+θ0=C−m−θ0$ from Fig. 1. Consequently, the field distributions in these two reciprocal cases are just opposite with each other with respect to the normal axis along x direction, that

$E_{\rm{ext}}^{+\theta _0}(r,+\phi )=E_{\rm{ext}}^{-\theta _0}(r,-\phi )$

$E ext +θ0(r,+ϕ)=E ext −θ0(r,−ϕ)$⁠. The diffraction coefficients of each orders, which are governed by the distribution of field in the array, are identical, i.e. the diffraction is reciprocal.

When the cylinder is formed by a MO medium, this reciprocal effect is broken. When a magnetic field H₀ is applied in the in-axial z direction, the time-reversal symmetry of an optical wave propagating inside the cylinder is broken, especially to PAMSs with transverse rotating energy fluxes in the x − y plane.¹⁸ From Figs. 1(a) and 1(b) we can see PAMSs with opposite m values should be excited in these cases of opposite incidence. However, according to the classic analogue of the Zeeman effect, the ±m PAMSs no longer possess degenerated resonant frequencies ω_m/2π and field distributions Ψ_m(r).¹⁸ Different sets of PAMSs with

$C+m+θ0≠C−m−θ0$ are excited in these two cases, leading to nonreciprocal diffraction effects with unequal transmission efficiencies or/and wavelengthes. The classic analogue of the Zeeman effect on PAMSs is the main mechanism of the possible observable optical non-reciprocality.

By using full-field three-dimensional finite element optical simulations (COMSOL Multiphysics 4.3a), we study the diffraction of an optical wave from a MO cylinder array by calculating the transmission coefficients of various diffraction orders. A schematic of the MO cylinder array under investigation is shown in Fig. 1. Radius of each MO cylinder is a = 1 cm, and period of the array is d = 2.9 cm. The array is aligned along the y direction, and the incident optical wave forms an angle of incidence θ with respect to the normal of the array in the −x direction. The incident wave is transverse electric (TE) polarized, i.e. with an electric field E_z along the z direction. This polarization can excite the degeneracy-broken PAMSs within our interest in this article.

In the coordinate frames of (x, y, z), the magnetic permeability tensor of the MO medium can be expressed as^15,18

where the values of tensor elements μ_r and μ_k are functions of the applied external magnetic field H₀.^15,18 Usually the tensor elements μ_r and μ_k are complex with proper dispersions.^15,18 Here, in order to emphasize the physical mechanism behind the nonreciprocal diffraction we neglect the dispersions in μ_r and μ_k, and assume that μ_r = 0.9 and μ_k = −0.1 throughout the frequency regime studied in this article. This assumption permits us to reveal the role of the geometric dispersion arisen from the periodically arranged MO cylinders ambiguity. The permittivity ε of the cylinder is 15.26.¹⁵ 

With above parameters we can directly calculate the eigenfrequencies ω/2π of different PAMSs (m, p) in a single MO cylinder,¹⁸ where p is the radial index. Within the frequency regime from 14 GHz to 14.4 GHz, the available PAMSs are (m, p) = (−8, 1), (−5, 2), (5, 2), and (8, 1) with eigenfrequencies of 14.044 GHz, 14.042 GHz, 14.189 GHz and 14.299 GHz, respectively.¹⁸ High-m PAMSs with m = ±5 and m = ±8 are involved in the possible nonreciprocal diffraction effect discussed in this article.¹⁵ 

Note that according to the conservation of parallel wavevector of

$k‖(n)=k0sinθ+n2π/d$¹⁵ for the nth diffraction order, this MO array is subwavelength only for frequency smaller than 10.34 GHz. Unlike that in,¹⁵ here the structure is no longer subwavelength from 14 GHz to 14.4 GHz. Below we will show that various nonreciprocal optical transmission effects can be still obtained, and that the simultaneous existence of different diffraction orders, even under a normal incidence, helps us to get a deeper insight into the role of PAMSs in the scattering of an optical wave from the MO cylinder array.

Figure 2 shows the transmission spectra of the 0th diffraction order T₀ versus the angle of incidence θ. It is interesting to note that the transmission of this 0th order is reciprocal, i.e. T₀(+θ₀) = T₀(−θ₀). In sharp contrast with that of the 0th order, the transmission spectra of other nonzero diffraction orders are nonreciprocal. Examples of the −1st and −2nd orders are shown in Fig. 3, where for opposite angles of incident ±θ₀ the values of transmission coefficient T_n are different.

From the transmission spectra shown in Figs. 2 and 3 we can observe two unique features. One feature is that the transmission coefficients T_n varies strongly with the angle of incidence θ around two characteristic frequencies of f₁ = 14.090 GHz and f₂ = 14.250 GHz. The other feature is that a dispersionless quasi-omnidirectional nonreciprocal optical transmission effect can be observed for the −2st diffraction order, as shown in Fig. 3(b). To be more explicitly, when the angle of incidence θ is negative, a high transmission of the −2st diffraction order takes place around f₂. When θ is positive, a similar high transmission of T₋₂ can be obtained, albeit at f₁. The positions of the resonant transmission peaks depend very weakly on the magnitude of the incident angle θ. Note that a transmission of the −2nd diffraction order exists only when the magnitude of incident angle θ is greater than a critical value of θ_c given by

$k‖(−2)=k0$⁠. At frequency f₁ the critical value of θ_c is given by 27.93°. At f₂, θ_c equals 26.87°.

To present above discussed features more clearly, in Fig. 4 we show the transmission spectra of the 0th, −1st and −2nd diffraction orders at θ = ±50° and ±30°, respectively. We can see the nonzero (0th) diffraction orders are indeed nonreciprocal (reciprocal). The transmission of the −2st diffraction order is very strong at either f₁ or f₂, although its width changes with the angle of incidence θ. The transmission curves near f₁ and f₂ are of Fano line-shape, implying that there exists an interference between a localized resonance and a collective oscillation. Below, let us investigate what a kind of role the PAMSs play to this nonreciprocal diffraction effect by analyzing the distributions of field and energy flux.

Paying attention to the two characteristic frequencies of f₁ and f₂, where various diffraction orders show noticeable nonreciprocal or reciprocal transmission features, we simulate and analyze the corresponding distributions of field and energy flux.

Figure 5 shows the distributions of field and transverse energy flux at f₁ and f₂ for the angles of incidence at ±50°. We can see in all cases a homogenously clockwise or counter-clockwise rotating energy flux is excited in each MO cylinder of the array. For example, at f₁ under all angles of incidence a clockwise rotating energy flux is excited, see Figs. 5(a) and 5(c). Consequently, the topological charge m of the dominant PAMS is negative. Furthermore, from the distribution of field Re{E_z} we can see the resonance at f₁ is dominated by the (m = −5, p = 2) PAMS, because Re{E_z} varies 5 times when it circles the center, and two rings of high field distributions can be observed along r direction. One ring is localized just at the boundary, while the other one, which is also the strongest one, is inside the cylinder. Similarly, for the frequency at f₂ a homogenous counter-clockwise energy flux is excited, which is dominated by the (m = +5, p = 2) PAMS. These results are in consistent with our former analytical calculation, that within the frequency regime from 14 GHz to 14.4 GHz the (m = ±5, p = 2) are available in a single MO cylinder.

From above simulation we can see the m = ±5 PAMSs dominate the excited field at both f₁ and f₂. The associated rotating energy flux can qualitatively explain some nonreciprocal diffraction effects shown in Fig. 4, especially the dispersionless quasi-omnidirectional optical transmission effect of −2nd orders at f₁ and f₂. For example, at f₁ the energy flux is clockwise, so the diffraction should favor the clockwise one at the emitting side of the array. Consequently, at the incidence of +50° the −2nd diffraction order is greatly favored. The parallel direction of incident wavevector k₀sinθ at the front side of the array also fits well with that of the clockwise rotating energy flux of the MO cylinder, which further enhances the diffraction effect. When the angle of incidence is −50°, from the same consideration the diffraction should favor the positive +2nd diffraction order. However, this order is evanescent because

$k‖(+2)>k0$ and its diffraction is forbidden. For the same reason, at frequency f₂ the energy flux in each MO cylinder is counter-clockwise, so the diffraction should favors the counter-clockwise one at the emitting side, and at the incidence of −50° the −2nd diffraction order is enhanced.

To get a deeper insight into the mechanism behind this non-reciprocity, especially the role of collective inter-cylinder interaction, we analyze the electric field component E_z by decomposing it into a linear combination of exp (−jmϕ). We pick up the excited field inside a MO cylinder within a ring of 0.5 cm < r <0.6 cm, Fourier transform it by E_z(r) = Σ_mD_mexp (−jmϕ), and normalize D_m. Although exp (−jmϕ) is not the eigenfunction of the m PAMS, the weight |D_m|² provide us information about how the collective interaction among different MO cylinders modifies the excitation of PAMSs. Two standard results are shown in Fig. 6. We can see at the characteristic frequency f₁, although m = −5 PAMS is always the dominant component, the collective interaction helps to excite the m = −9, −1, and +5 PAMSs, even when their eigenfrequencies in a single MO cylinder are far away from f₁. Under different angles of incidence the weights of each PAMS are different, especially that of the m = +5 PAMS. Similar effect can be found at the other characteristic frequency of f₂, where the m = +5 PAMS is the dominant one, and the collective interaction excites the m = +9, +1, and −5 PAMSs.

Normal incidence is an interesting situation that deserves our attention. Especially, with the geometric parameter utilized in this article, under normal incidence the transmitted wave contains the ±1st diffraction orders.

The calculated transmission spectra of the 0th and ±1st diffraction orders are shown in Fig. 7. We can see the transmission coefficients of the ±1st diffraction orders are not equal. At frequency f₁ the +1st diffraction order is the dominant one and the −1st diffraction order is quenched. At f₂ the reversed result can be obtained, and the diffraction favors the −1st direction. Similar to those shown in Fig. 5, the transverse energy flux at these two characteristics frequencies is also clockwise and count-clockwise, respectively, as shown in Fig. 8.

From above simulations we can see the transmission through the MO cylinder array is dominated by the m = ±5 PAMSs. However, from the analysis about PAMSs in a single MO cylinder we realize that from 14 GHz to 14.4 GHz the m = ±8 PAMSs are also supported. It is then interesting to see where these m = ±8 PAMSs dominate the optical diffraction effect.

By analyzing the transmission spectra and the distribution of field at different frequencies, we find that the m = ±8 PAMSs play an important role in the two spikes at 14.104 GHz and 14.359 GHz. These two spikes can be clearly observed from the transmission spectra, for example, in Figs. 4(c) and 4(d), where the transmission curves change sharply. The associated distributions of field and transverse energy flux are shown in Fig. 9, where a strong excited field with an eight-fold symmetry can be observed. The field pattern clearly shows that the associated radial index p is 1, in consistent with our analytical evaluation before for PAMSs in a single MO cylinder.

From the simulation results about the distributions of excited field, we can see that in sharp contrast with those shown in Fig. 5, now the fields in adjacent MO cylinders vary strongly. It is because a Floquet periodicity condition is utilized in the COMSOL simulation for this oblique incidence, and the field Re{E_z} is the instant real part of the harmonic-oscillating electric field. We also analyze the m spectrum. Two standard results at 14.104 GHz are shown in Fig. 10. We can see although the m = 5 and 4 are excited as well, the m = 8 PAMSs dominates the collective oscillation in the MO cylinder array at 14.104 GHz. The weight of each m PAMS also depends on the angle of incidence θ.

In above sections we demonstrate that nonreciprocal optical diffraction can be obtained via exciting the collective oscillation of PAMSs, which possess split frequencies due to the classic analogue of the Zeeman effect. It proves that degeneracy-broken PAMSs play important roles in many nonreciprocal effects, e.g. the dispersionless quasi-omnidirectional nonreciprocal diffraction of the −2nd order is correlated to the resonant excitation of rotating energy flux in the MO cylinder array. Some other interesting phenomena are found from the full-wave COMSOL simulation, i.e. the reciprocal diffraction of the 0th diffraction order, and the weak excitation of the m = ±8 PAMSs at the spikes. Before ending this article, in this section we would like to provide a further discussion and analysis about some characters that need our attention.

First, the 0th diffraction order is reciprocal and does not depend on the sign of the incident angle. This effect can be easily explained by realizing that the excitation of PAMSs by an incident wave and the radiation of the 0th diffraction order from the excited PAMSs are complementary with each other. As schematic shown in Fig. 11, although for opposite angles of incidence different sets of PAMSs are required to compensate for the phase difference ΔΘ, for the emitting process of the 0th diffraction order the exactly complementary and inverted process takes place. To be more explicitly, if at the front side of the array a clockwise (counter-clockwise) energy flux S is needed, then at the rear side of the array a counter-clockwise (clockwise) energy flux S_a should be a necessary to compensate for the inverted phase shift −ΔΘ between adjacent cylinders. If the whole process of the 0th-order optical transmission through the cylinder array can be equivalent to the transmission through a homogenous medium with two different mirrors M_L and M_R, where L and R represent the direction of the rotating energy flux shown in Fig. 11, the diffraction of the 0th order at opposite angles of incidence is just equivalent to switching the order of the two equivalent mirrors M_L and M_R, as schematic illustrated in Figs. 11(c) and 11(d). The two cases would then give exactly the same transmission coefficient T₀. Note that this explanation could not apply to the nonzero nth diffraction order because at the emitting side the phase shift is −ΔΘ − nπ other than the complementary one of −ΔΘ. Nonreciprocal diffractions are then expected for these nonzero orders.

Second, from the analysis of m spectra we can see the transmission in the frequency regime from 14 GHz to 14.4 GHz is dominant by the m = ±5 PAMSs. Although the resonant frequencies of the m = ±8 PAMSs are also within this frequency regime of investigation, they contribute very little to the resonant transmission. This effect can be briefly understood from the fact that the field of a PAMS with a smaller |m| value varies more slowly along the cylinder boundary. Larger scattering cross-sections are then expected for the m = ±5 PAMSs by comparing with those of the m = ±8 PAMSs.^19–21 

Above statement is also verified by analyzing the variations of characteristic transmission frequencies versus the angle of incidence θ and the period d. The spikes associated with the m = ±8 PAMSs hardly vary with θ and d, while the two bands of dispersionless quasi-omnidirectional optical transmission are subjected to observable small shifts. It implies that the optical resonances of the spikes are of a stronger localized nature²² than these of the dispersionless quasi-omnidirectional optical transmission. This strong localized nature makes it difficult to excite these m = ±8 PAMSs, and once excited, the transmission coefficients change more sharply than these peaks dominated by the m = ±5 PAMSs. Note that for a resonant optical transmission through a grating by exciting a strongly localized mode, the transmission frequency should coincide with the eigenfrequency of this mode in a single cylinder. Above simulations show that there are differences between the eigenfrequencies of the m = ±8 PAMSs (14.044 GHz and 14.299 GHz) from the analytical calculation¹⁸ and the transmission frequencies of spikes from the full-wave simulation (14.104 GHz and 14.359 GHz). The 0.4% frequency difference might be attributed to the system error associated with the grid division in COMSOL.

Third, from the m-spectra shown in Figs. 6 and 10 we can observe an interesting phenomenon. From Fig. 6 we can see the dominant modes are these PAMSs with odd m values, i.e. m = ±5 and m = ±9. This phenomenon is in consistent with the broken rotation symmetry of each MO cylinder by the arrangement of the array along y direction, which, in fact, introduces a two-fold rotation symmetry. Consequently, the excitation of m = ±5 PAMSs also renders the excitation of some other odd-m PAMSs. On the other hand, for these spikes where the dominant modes are PAMSs with even m values besides the m = ±5 ones, a similar effect can be obtained, i.e. some other even-m components are excited. The excitation of m = ±5 PAMSs at spikes can be understood as a background contribution.

Last, although most characteristics of the optical diffraction from the MO cylinder array can be explained by the classic analogue of the Zeeman effect on PAMSs and the excitation of rotating transverse energy flux, it could not qualitatively explain the nonreciprocal diffraction of some diffraction orders, especially the −1st order. To fully explain the nonreciprocal features of the −1st diffraction order, analysis from the eigenmodes in the MO cylinder array by taking into account of proper interference effects is required.^23,24 However, at present we could not develop a simple method in finding the eigenfrequencies and eigenfunctions of the eigenmodes confined to this MO cylinder array, especially because the optical response of the MO medium is anisotropic.¹⁵ 

In summary, we provide a comprehensive study on the nonreciprocal optical diffraction of various orders from a MO cylinder array. We show that nonreciprocal diffraction can be obtained only for these nonzero orders, and the 0th diffraction is reciprocal. Some interesting phenomena such as the dispersionless quasi-omnidirectional nonreciprocal diffraction and spikes associated with high-m PAMSs are present. Mechanism behind these nonreciprocal effects are discussed. This investigation verifies that PAMSs play an important role in manipulating the diffraction of optical waves from cylinder arrays. Because the off-diagonal element μ_k is tunable by changing the magnitude and direction of the external applied magnetic field H₀ in z direction,^15,18 the interplay between the inter-cylinder coupling in y direction and the classic analogue of the Zeeman effect on PAMSs makes its feasible to design many flexible nonreciprocal optical devices for future on-chip integration demands.

The authors acknowledge the support from the National Natural Science Foundation of China (NSFC) under grant 11174157, and the Specialized Research Fund for the Doctoral Program (SRFDP) under grant 20110031110005.