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We study the diffraction of optical waves by a single layer of gyromagnetic cylinders. We show that a nonvanishing rotating dipole momentum is excited in a single gyromagnetic cylinder because of the classic analog of the Zeeman effect on photonic angular momentum states (PAMSs). Consequently, different collective dipole modes are excited in a gyromagnetic cylinder array at opposite incident angles. Nonreciprocal optical diffraction effects can be observed, where the transmission and reflection coefficients depend on the sign of the incident angle. A novel phenomenon of nonreciprocal negative directional transmission is demonstrated and numerically analyzed. This work highlights the potential of PAMSs in manipulating the propagation of optical waves for various applications.
© 2014 Optical Society of America
1. Introduction
Nonreciprocal optical effects have become a popular research topic because of their wide range of applications [1]. For example, optical isolators could be utilized to prevent undesirable parasitic feedback in an optical circuit. These optical isolators forbid the backward propagation of an optical mode by nonreciprocal refraction or absorption from Faraday effect in magneto-optic (MO) media [2]. Nonlinear effects can be utilized for many nonreciprocal applications [1], especially in realizing optical diodes characterized by an asymmetric transmission [3, 4]. Considering the growing interest in meeting the demand for on-chip applications and the rapid advances of metamaterials and photonic crystals, many new nonreciprocal mechanisms and phenomena in optical nanostructures have been proposed and demonstrated over the past decade [5–17]. Nonreciprocality in the constituent artificial resonant units made of MO media with asymmetric dielectric constants remains the most frequently utilized mechanism [5–12].
Each constituent element in a metamaterial or a photonic crystal supports various resonant eigenstates. For a cylinder or coaxial element, the optical eigenstates are photonic angular momentum states (PAMSs) that are characterized by the topological charge m in the helical phase distribution of exp(− jmφ), where φ is the azimuthal angle [18–21]. The unique characteristics of PAMSs in metamaterials can be utilized for various functions and applications, see [21] and the works referenced therein. In terms of nonreciprocal properties, the resonant frequencies of PAMSs with opposite m can be modified with varying degrees by using the MO effect to break the time-reversal symmetry (TRS). This phenomenon is called the classic analog of the Zeeman effect [18,21]. This frequency splitting effect provides some unusual decoupling properties that could be useful for designing novel photonic devices [18]. However, explicit discussions on the feasibility of utilizing this effect to realize nonreciprocal optical diffraction are lacking in the literature.
In this article, we study the optical diffraction from subwavelength MO cylinders. Using gyromagnetic yttrium-iron garnet (YIG) as an example, we show that a rotating dipolar momentum can be obtained in a single YIG cylinder because of the classic analog of the Zeeman effect on PAMSs. The excited field is no longer symmetric with respect to the incident wavevector k. Consequently, the collective dipole mode excited in a single-layer array of YIG cylinders is sensitive to the sign of the incident angle θ, and the diffraction of an optical beam becomes nonreciprocal. We also demonstrate nonreciprocal negative directional transmission. When the incident optical beam propagates along a proper angle of incidence −θ0, a total transmission is achieved, and the direction of the transmitted beam is along that of the negative direction, i.e., at the same side of the normal n as that of the incident beam. However, for the incidence at +θ0 the optical beam is totally reflected. Numerical analysis proves that PAMSs with the classic analog of the Zeeman effect have important functions in this nonreciprocal diffraction phenomenon. This investigation shows that PAMSs in nanostructures can be manipulated via the MO effect to realize various nonreciprocal diffraction effects, and can find various applications in different disciplines.
This article is organized as follows. In subsection 2.1, we briefly discuss how the classic analog of the Zeeman effect on PAMSs modifies the distribution of the excited field in a single YIG cylinder. In subsection 2.2 we pay attention to the optical diffraction from a single-layer array of YIG cylinders. We show that different collective dipole modes are excited when the incident beam reverses its incident angle. The diffraction coefficients now depend not only on the magnitude of the incident angle θ, but also on its sign. The phenomenon of nonreciprocal negative directional transmission is demonstrated and analyzed. In subsection 2.3 the dependence of the nonreciprocal diffraction on the magnitude of the external static field H0 is discussed. Further comments on the possible analytical analysis of this nonreciprocal diffraction and the feasibility in realizing this effect in other frequency regimes are provided in subsection 2.4. Summary is made in Section 3.
2. Simulation and analysis
2.1. In a single YIG cylinder
We investigate how the MO effect influences the optical field in a single YIG cylinder via the classic analog of the Zeeman effect on PAMSs. A static magnetic field H0 is assumed to be applied in the axial +z direction of a YIG cylinder, see Fig. 1. The magnetic permeability tensor of YIG can be expressed as [11, 18]
where ωk = γH0 is the precession frequency, γ is the gyromagnetic ratio, ωm = 4πγMs, and 4πMs is the saturation magnetization. From the physically accessible material parameters of H0 =800 Oe and 4πMs =1884 G [11], the tensor elements are μr = 0.9952 and μk = −0.1064 at frequency f0 = 7.91 GHz, respectively. The permittivity ε of YIG equals 15.26 [11].
Fig. 1 Distributions of electric field Re{Ez} when (a) μk = 0, and (b) μk = −0.1064, respectively. Plot (c) shows the definition of φB between k and the axis p representing the asymmetric distribution of Ez.
The off-diagonal element μk in the permeability tensor determines TRS of the medium [11, 18]. When μk = 0, TRS is preserved. Using the finite-element method (COMSOL Multiphysics 4.3a) we study the scattering of a transverse electric field by a cylinder with μk = 0, where the radius r of the cylinder equals 1 cm. As shown in Fig. 1(a), the field profile Ez is symmetric with respect to the direction of incidence k that points along the +x direction. The azimuthal angle φ is the angle formed with respect to the +x axis. Hence, the symmetric field profile of Ez(+φ) = Ez(−φ) implies that the +m and −m PAMSs are excited equally, i.e., e+m = e−m, where Ez(φ) = ∑em exp(− jmφ). This statement is proven by analyzing the m spectrum inside the cylinder [19–21], which shows that weights of the dominant components are |e±1|2=16.0%, |e±2|2=18.8%, and |e±4|2=9.7%. Evidently, the classic analog of the Zeeman effect on PAMSs is absent here.
When μk ≠ 0, TRS in YIG is broken and the classic analog of the Zeeman effect lifts the degeneracy between the +m and −m PAMSs, especially in their excitation efficiencies, i.e., e+m ≠ e−m. Figure 1(b) shows the result when μk = −0.1064. As expected, now the electric field is asymmetrically distributed with respect to k, i.e., Ez(+φ)≠ Ez(−φ). A counterclockwise vortex of time-averaged energy flux can be observed inside the YIG cylinder. Although the whole field profile of Ez excited in the YIG cylinder is generally asymmetric and does not possess any reflectional symmetric axis, to represent the averaged distribution of field we can still define an axis p, which forms an equivalent inclined angle of φB > 0 with respect to k [see Fig. 1(c)]. We also analyze the m spectrum of the excited field [18–21]. In sharp contrast to the case of μk = 0 shown in Fig. 1(a), the weight of the m = −1 PAMS can reach |e−1|2 =93.4%, which is significantly greater than that of the m = +1 PAMS. Evidently, PAMSs with opposite topological charges of +m and −m are unequally excited because of the broken degeneracy from the classic analog of the Zeeman effect, and their superposition contributes to the nonvanishing rotating energy flux [18–21] and the asymmetric field distribution represented by p. A similar effect on transverse energy flux is also obtained for chiral Hall edge states in gyrotropic photonic clusters [13].
2.2. Nonreciprocal diffraction from a YIG cylinder array
The results in the previous subsection show that the classic analog of the Zeeman effect leads to an unequaled excitation of PAMSs with opposite m and that the excited field inside the YIG cylinder is not symmetric with respect to k. This characteristic is the key to many potential incident-angle-dependent nonreciprocal diffraction effects from a single-layer array of MO cylinders.
To obtain more insights, we consider the schematic configurations shown in Fig. 2, and neglect the inter-particle coupling. When the angle of incidence θ is −θ0, the orientation of p in each YIG cylinder is along the direction of ϕ− = φB − θ0 with respect to the normal n [Fig. 2(a)]. However, when the wave is incident from the reverse direction, i.e., at an incident angle θ of +θ0, p in each YIG cylinder is not reversed to the direction of −ϕ−. In sharp contrast to Fig. 2(a), Fig. 2(b) shows that p forms an angle of ϕ+ = φB + θ0 from n. With the reflectional symmetry of the single cylinder array with respect to n, these two configurations are equivalent only when ϕ+ + ϕ− = 0 or π. Consequently, if φB ≠ 0 or π/2, the spatial distributions of p at these two opposite angles of incidence are no longer degenerated. When inter-particle coupling is considered, the collective dipole modes excited in the single array of cylinders would determine the weights of various diffractive orders in an optical scattering process [22, 23]. The spatial orientations of p schematically shown in Fig. 2, which would form different collective dipole modes, are expected to produce nonreciprocal optical diffraction.
Fig. 2 Orientations of p without inter-particle coupling when the angle of incidence θ equals (a) −θ0 and (b) +θ0, respectively. With the consideration of inter-particle coupling, different collective dipole modes are expected in these two configurations, which lead to nonreciprocal diffraction.
Note that the nonreciprocal effect discussed in this study does not modify the propagation angles of the diffracted modes. The propagation direction of the lth diffractive order is still governed by the conservation of parallel wavevector of
where d is the period of the array and k0 = 2π/λ. The nonreciprocal effect modifies the transmission or reflection coefficients of these diffractive orders, which depend on the magnitude and sign of the incident angle θ.We study the optical diffraction from a single-layer array of YIG cylinders with a period d of 2.9 cm to verify the simple intuitive consideration described above. The transmission coefficients T0 and T−1 of the 0th and −1st diffractive orders for μk = 0 and μk = −0.1064 are displayed in Fig. 3. The magnitude of should be smaller than that of k0, i.e., , to ensure that the lth diffractive order is radiative. Consequently, the −1st diffractive order at f0 = 7.91 GHz is observable when the magnitude of the incident angle |θ| > 17.93°. Other higher diffractive orders are evanescent for all values of θ and cannot be observed in the far field.
Fig. 3 Transmission coefficients T of the 0th (blue) and −1st (red) orders versus the angle of incidence θ, when μk = 0 (short dashed) and μk = −0.1064 (solid), respectively.
As shown in Fig. 3, the diffraction is reciprocal, i.e., T0,−1(+θ0) = T0,−1(−θ0), when μk = 0. The transmission coefficient of the 0th diffractive order T0 is 48.4% when θ is zero. T0 decreases monotonously as the angle of incidence becomes larger. The −1st diffractive order can be observed when θ is greater than 17.93°. T−1 is greater than T0 when θ > 20°, and the maximum T−1 can reach 18.7% when θ is approximately 40°.
When TRS of the medium is broken by using a nonzero value of μk = −0.1064, the diffraction characteristics are greatly modified. For example, the transmission coefficient T0 is reduced to 20.1% at normal incidence. The most pronounced feature is that the transmission coefficient of the −1st diffractive order evidently depends on the sign of θ, i.e., T−1(+θ0) ≠ T−1(−θ0). This nonreciprocal feature is especially strong at θ0 = 40°. When θ = −40°, T−1 is greatly enhanced and approaches 91.1%. When θ = +40°, T−1 is reduced to 2.9%. As a consequence of the boosted diffraction of the −1st order at θ = −40°, the reflection coefficient R is reduced to 5.9% from the initial value of 75.7% at μk = 0. By contrast, when θ = +40° the value of R is enhanced to 95.3%. We can see the reflection coefficient is also nonreciprocal.
Figure 3 implies the existence of a novel nonreciprocal phenomenon, namely, nonreciprocal directional negative transmission. When θ = −40° and +40°, a nearly perfect negative directional transmission and a nearly total reflection is obtained, respectively. Although each diffraction behavior has been demonstrated in isotropic dielectric cylinder arrays [22, 23], a simultaneous realization of these two effects in a single structure has not been reported in any literature, to the best of our knowledge. We show this nonreciprocal phenomenon in Fig. 4 by modeling the incidence of an optical beam to the YIG cylinder array. Figures 4(a) and 4(b) evidently show this nonreciprocal diffraction effect. For comparison, in Fig. 4(c) we also show the effect observed when the off-diagonal element μk is set to zero.
Fig. 4 Optical diffraction by a single-layer array of YIG cylinders, for nonreciprocal cases of μk = −0.1064 at (a) θ = −40° and (b) θ = +40°, respectively, and for reciprocal case of μk = 0 at (c) θ = +40°.
We simulate and analyze the corresponding distributions of field and energy flux to emphasize that the different collective dipole modes excited in the YIG cylinder array contribute to this nonreciprocal diffraction effect. Results are shown in Fig. 5.
Fig. 5 Distributions of electric field Re{Ez} and energy flux (by arrows) at the central three YIG cylinders for nonreciprocal cases of μk = −0.1064 at (a) θ = −40° and (b) θ = +40°, respectively, and for reciprocal case of μk = 0 at (c) θ = +40°.
As shown in Fig. 5(a), a homogenous counterclockwise energy flux is excited in each YIG cylinder of the array when the angle of incidence θ is −40°. Albeit the profile of field Ez in each cylinder is similar to that shown in Fig. 1(b) and is dipolar symmetric, its orientation becomes normal to k because of the inter-particle coupling. Moreover, the field in adjacent cylinders has a phase difference of π [22]. Thus this collective resonance contains both transverse and longitudinal components [23]. At the outgoing side, the fields from a pair of adjacent cylinders interfere to produce radiation mainly in the 0th and −1st diffractive orders. However, the scattered field of the 0th diffractive order possesses a phase shift of π with respect to the incident field because of the dipolar symmetry of the field in each single YIG cylinder, and they cancel each other out [22]. Consequently, only the −1st diffractive order is left to propagate away from the YIG array, exactly along the direction of +40°.
A total reflection takes place when θ = +40°. Compared with the field distribution shown in Fig. 5(a), that in Fig. 5(b) illustrates that a different collective mode is excited in the YIG array. The field profile in each single YIG, albeit weak, is reflectional symmetric with respect to n, and reflectionally antisymmetric with respect to the center-to-center chain axis. The excited field in adjacent cylinders also possess a phase difference of π. These properties enable the excited collective modes inside the YIG array to decompose into two grating modes of the chain, one being longitudinal and the other being transverse [23]. These modes contribute to two corresponding transmitted wave components that cancel each other out, thereby quenching the transmitted diffractive orders and completely reflecting the incident wave [23].
We analyze the m spectrum in a single YIG cylinder of the array to determine whether or not PAMSs dominate this nonreciprocal diffraction effect. When θ = −40°, the m = −1 PAMS still dominates the resonance inside the YIG cylinder, and its weight |e−1|2 can reach 95.7%. However, when θ = +40° its weight |e−1|2 is reduced to 29.8%. Some other PAMSs are excited, especially the m = +3 and +2 PAMSs, which weight 48.7% and 13.6%, respectively. This analysis shows that the excitation of PAMSs is modified when collective modes are excited inside the YIG array via the inter-particle interaction. The m spectra of the collective modes are sensitive to the sign of θ, which contributes to the nonreciprocal optical diffraction.
For comparison, we also simulate the case when the off-diagonal elements are zero. The result is shown in Fig. 5(c), where, according to Fig. 3, the transmission coefficients of the 0th and −1st orders are 5.6% and 18.7%, respectively. These values, albeit weak, are still greater than the T−1 =2.9% of the nonreciprocal case shown in Figs. 4(b) and 5(b). The distributions of field and energy flux are different from the two nonreciprocal cases shown in Figs. 5(a) and 5(b), and a noticeable resonance with nearly dipolar symmetry with respect to n is excited in each cylinder.
The difference in the collective modes excited in the single-layer array of YIG cylinders due to the classic analog of the Zeeman effect on PAMSs is evidently the key mechanism of the nonreciprocal diffraction effect. However, we only consider the situation where the transmission coefficient T−1 of the −1st diffractive order is greatly nonreciprocally modified. It might be worth determining what happens when only one diffraction mode, the 0th one, is transmitted. Special attention should be paid to the case of normal incidence because now the incidences of +θ0 and −θ0 are degenerated.
The simulation results for both cases of μk = 0 and μk = −0.1064 are shown in Fig. 6. The transmission coefficients T0 in these two cases are 48.4% and 20.1%, respectively. The distribution of field shows that the excited field in each cylinder occurs along the normal direction of n when μk = 0 [see Fig. 6(a)]. However, when TRS is broken with μk = −0.1064, a strong energy flux and an inclined field distribution from n can be observed, i.e., the collective resonance is not symmetric with respect to the center-to-center chain axis [see Fig. 6(b)]. As for the m spectra in these two cases, the +m and −m PAMSs are excited equally in Fig. 6(a). The weights of the ±1 and ±4 PAMSs are 25.2% and 16.1%, respectively. For the case shown in Fig. 6(b), the dominant mode is the m = −1 PAMS with a weight of 94.3%. The classic analog of the Zeeman effect on PAMSs still has an influence on this case of normal incidence.
Fig. 6 Distributions of electric field Re{Ez} and energy flux at normal incidence, when (a) μk = 0, and (b) μk = −0.1064, respectively.
2.3. Dependence on H0
According to Eqs. (1) to (3), the MO response of YIG is a function of the external static magnetic field H0. Consequently, the performance of the nonreciprocal diffraction effect depends on H0. We briefly study the variation of the nonreciprocal diffraction effect as a function of H0, as shown in Fig. 7.
Fig. 7 Diffraction coefficients versus frequency at different external applied magnetic field H0 for (a) the −1st diffractive order when θ = −40°, and (b) the reflected beam when θ = +40°, respectively.
When θ = −40°, two high transmission peaks of the −1st diffractive order can be obtained, and these peaks move to the higher frequency regime when H0 increases. When H0 =800 Oe, one peak is localized at f0 = 7.91 GHz, and the other is localized at fh = 8.55 GHz. By contrast, a high reflection band with a width greater than 1 GHz, which covers the frequency regime from f0 to fh, can be obtained when θ = +40°, The nonreciprocal properties at fh, which are dominated by the m = −2 PAMS, are also studied (not shown here). The same mechanism of f0 discussed in the previous subsections can be applied to fh as well.
2.4. Further discussion
Our numerical simulations and analyzes prove that the collective mode in a single-layer array of YIG cylinders depends strongly on the sign of the incident angle θ. This nonreciprocal effect can lead to various interesting diffraction phenomena, especially the nonreciprocal directional negative transmission. The characteristics of this nonreciprocal effect from first principles [22–25] should be further studied to understand the relevant physical mechanism. However, hybridization of the longitudinal and transverse resonances, dispersion during the inter-particle interaction, the complex role of evanescent waves, and the anisotropic optical response of the MO medium make the analytical study on this topic an interesting but intricate subject of investigation.
In this study, we focus on YIG, which has a strong MO response in the GHz regime. The proposed effect and mechanism apply equally to other optically active media with off-diagonal permittivity elements [5–9, 21, 26] based on the principle of electromagnetic duality. The realization of this kind of nonreciprocal effect at high frequency, e.g., THz and infrared, is also possible. For example, we can utilize a novel material with proper values of charge density n, effective charge e* and mass m* to realize optical activity at high frequency regime [21, 26]. Nanostructures have already been shown to have the ability to enhance the MO effects, and might help to reduce the size of photonic devices [27, 28] for on-chip applications [29]. With the advances of metamaterials and state-of-the-art fabrication techniques, novel ways in realizing artificial strong optical activity are also expected, see [7, 30] and references therein.
With the infinite length of the cylinder in the z direction, the nonreciprocal effect discussed in this study is actually two-dimensional. Cylinders are idea elements that support PAMSs. However, in realistic configurations and applications more realistic practical three-dimensional geometries should be employed [7], including ellipsoids [10, 12], truncated rods [11] and flat disks [12]. The mechanism discussed in the study can be applied as well if the field distribution in each single element of the three-dimensional geometries becomes asymmetric with respect to k when an external applied magnetic field is applied. Possible realization by using surface plasmon polaritons needs our special attention because it might push forwardly the advances of realistic on-chip applications.
3. Conclusion
In summary, we study the diffraction of optical waves by a single-layer array of YIG cylinders. We show that due to the classic analog of the Zeeman effect on PAMSs, at opposite angles of incidence different collective modes are excited in the YIG cylinder array. As a consequence, nonreciprocal diffraction effects can be observed, where the transmission and reflection coefficients depend not only on the magnitude of the incident angle, but also on its sign. A novel effect of nonreciprocal negative directional transmission is demonstrated. Numerical simulation and analysis prove the important role played by PAMSs with lifted degeneracy from the classic analog of the Zeeman effect. Performance of this nonreciprocal effect can be tuned by changing the applied external magnetic field H0. This investigation provides insights into the importance and various applications of PAMSs in different disciplines, especially in designing compact optical components for future on-chip applications [29] by utilizing the subwavelength size of the cylinders.
Acknowledgments
The authors acknowledge the support from the National Natural Science Foundation of China (NSFC) under grants 11174157, 11074131 and 11274183, and the Specialized Research Fund for the Doctoral Program (SRFDP) under grant 20110031110005.
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