In order to construct flat-top magneto-optical isolators (MOIs), we have performed a theoretical study on the case of transmission-type one-dimensional magnetophotonic crystals (MPCs). We have introduced high performance MPC structures with flat-top responses and with the capability of adjusting to perfect MOIs. The adjustment is carried out by tuning the applied magnetic field. All introduced MOIs are sufficiently thin with acceptable transmission bandwidth. In the best case, we have achieved a 19.42 μm-thick perfect MOI with the flat-top width of 7.2 nm. For practical purposes, we have also considered the influence of the error in thickness of individual layers on the operational parameters of the MOIs and investigated the possibility of compensating the deviations by the magnetic adjustment.
© 2012 OSA
Optical isolator is an inevitable element in telecommunication systems. This nonreciprocal device eliminates unwanted back-reflections that typically create instabilities such as mode hop, amplitude modulation, power spikes, and frequency shifts. These instabilities can permanently damage equipments such as lasers. Optical isolators generally, consist of two linear polarizers with polarization axes offset equal to 45°, and a non-reciprocal polarization-rotation element placed between the polarizers. The resultant device can produce a 90° polarization rotation for light in a round-trip. In order to access integrable optical isolators, many techniques and designs such as nonreciprocal micro-ring resonators [1, 2], nonlinear couplers [3, 4], photonic crystals (PCs) [5, 6], nonreciprocal waveguides [7, 8], nonreciprocal optical resonators on silicon- or Ce:YIG/Silicon-on-insulator substrates [9, 10], devices based on nonreciprocal phase shifts , photonic transitions [12, 13], surface plasmons [14, 15] and Mach-Zehnder interferometers [16, 17] have been employed.
One of the best candidates for integrated isolators are the magneto-optical (MO) isolators based on magnetophotonic crystals (MPCs) [18–25]. When a PC contains magnetic substances an MPC is constructed. In these structures, existence of photonic band gaps (PBGs) can be employed to enhance the MO effects, such as Faraday and MO Kerr responses. Because of PBG, certain wavelengths of light are strongly reflected from such structures. However, in presence of defect layers, transmission resonances are created within the gap, which allow some lightwaves with previously forbidden wavelengths to propagate inside the structure.
There are some theoretical works indicating possibility of designing magneto-optical isolators (MOIs) using 1-D MPC structures having multiple defects. These MOIs suffer either from narrow bandwidths [18, 22, 25] or from large thicknesses in spite of having flat-top bandwidths . Devices built upon MPC structures with bandwidths only in a fraction of a nanome-ter at central wavelengths, would encounter instabilities due to the wavelength fluctuations. In this paper, we introduce adjustable thin MPCs containing only few number of magnetic defect layers that are capable of providing the exact characteristics of T=100% and θF =45° for perfect MOIs and have a sufficiently flat-top bandwidth for stable operation. The functioning of MPC-based MOIs as devices operating at certain conditions (θF =45°) may be deviated by fabrication errors. Therefore, we will also study the stability of our introduced flat-top MOIs against minor thickness errors of their individual layers and show how the adjustment technique can compensate the influence of construction inaccuracies.
To introduce proper adjustable MPCs we have used a 4×4 transfer matrix method (TMM). In compare to this numerical method, an analytical approach may provide a better insight about the correlation between different components of the system under consideration, but in contrast, the results of TMM are exact, it is not restricted to especial cases and no important simplifying assumption is employed into the method. The TMM utilizes two fundamental 4×4 matrices accounting for propagation of light through each layer and boundary conditions at interfaces in the structure . To explain this method, we consider a beam of light impinging on the top of the multilayer from the air. If Pi is the electric field at the bottom surface in the ith layer, then a relation between electric field in air, Pa, and that in semi-infinite substrate, Ps, can be given by27]: Eq. (1) as: Eq. (6), for simplicity we have assumed εxx=εyy=εzz≡ε1. Furthermore, regarding the operational wavelength range, the relative permeability is taken equal to unity.
MPCs that are able to operate at transmission-mode are indispensable for fulfilling the MOI characteristics, provided that they can represent the appropriate optical and MO properties of T=100% and θF =45°. In order to design perfect MOIs, firstly, we introduce high performance MPC structures with Faraday rotations larger than what is required for MOIs. Then, we will adapt them to be specific for MOIs. We do this adaption by adjusting the gyration parameter of the magnetic media . This adjustment can be possible in Ce:YIG because its gyration parameter is proportional to its magnetization which can be changed by the external magnetic field. It is known that the strength of the magnetostatic interaction between the individual magnetic layers can be changed, and that a system of magnetostatically related films represents some increase in the saturation magnetic field compared with single layer films . However, we can neglect this because our procedure is carried out as an external adjustment in which desired response will be obtained by varying the applied magnetic field.
We may consider the applied magnetic field to be directed along the normal to the films where decreasing of magnetic field can alter the gyration parameter from its maximum at saturation magnetization to a lower value. For a magnetic garnet thin film with a single-axis anisotropy perpendicular to the surface of the film, reduction of normal magnetic field from saturation may lead to appearance of magnetic domains. For a typical domain size less than 5 μm , the incident light with a spot size of ∼2 mm comes up against an ensemble of antiparallel magnetic domains in magnetic garnet thin layers. In such a case, the magnetic layer can be substituted with an equivalent single-domain magnetic film whose magnetization vector is the statistical summation of magnetization in various domains. Consequently, when the spot size is very larger than the domain width, the dielectric tensor of the magnetic medium is in the form of Eq. (6) and thus the operation of the adjustable MOIs will be reliable. However, in some technical situations we may have to make use of a light beam with relatively small spot size. On the other hand, for uniaxial garnet films with a large perpendicular anisotropy field, controlling of gyrotropy with the normal external field will be difficult. For these situations, applying an oblique saturation magnetic field is recommended for which the magnitude of magnetization vector is preserved and a longitudinal component of magnetization comes to existence. In fact, the adjustment is carried out by rotation of magnetization vector.
In the presence of a longitudinal component of magnetization, another off-diagonal matrix element will emerge in the dielectric tensor of the magnetic layer [18,30]. When the magnetization vector is rotated in y–z plane, the gyration parameter, ε2, acquires components proportional to polar and longitudinal components of magnetization, as ε2xy ∝ M|| and ε2xz ∝ M⊥. Therefore, the permittivity tensor will get the form:
3. Results and discussion
For construction of MPCs, we utilize Si and SiO as dielectric layers and Ce:YIG as magnetic layer. During the past few years, Ce-modified yttrium iron garnet Ce:YIG has been increasingly used in various MO applications particularly in magnetophotonic crystal-based devices [24,31–35]. The growing interest toward the epitaxial growth of these materials in the form of thin films or etched films, for applications in integrated optics stems from their high Faraday rotation and good optical transparency in the visible and in the infrared spectral region, together with their compatibility with waveguide optics, low propagation losses, low saturation magnetic fields, and low production costs [36, 37]. The magnetic Ce:YIG layer has dielectric tensor elements ε1=4.884 and ε2=9×10−3 at λ =1550 nm . We consider its thickness to be λ/2nCe:YIG where, nCe:YIG represents the refractive index of Ce:YIG layer. At this optical communication wavelength, Si and SiO layers have refraction indices nSi=3.48  and nSiO=1.85 , and their thicknesses are set to λ/4nSi and λ/4nSiO, respectively. Hereafter, we denote SiO, Si and Ce:YIG layers as L (low index), H (high index) and M (magnetic), respectively. In some MPCs we may also utilize SiO2 or gadolinium gallium garnet (GGG) instead of SiO as L layer.
We are going to introduce adjustable MO isolators based on multi-cavity MPCs which are thin enough and show an acceptable transmission bandwidth. Cavity-type MPCs can be constructed by incorporating magnetic layers into dielectric PBG structures. They are capable of providing unique optical and MO characteristics by exploiting properties of band gaps and defects [34, 40–45]. It should be noticed that the bandwidth may be enhanced by increasing the number of defects in MPC structures . Therefore, firstly, we introduce several multi-cavity MPC structures as basic ones and then based on them, we obtain structures with flat-top responses. The basic structures are as follows:
- S(1) ≡(H/L)3/M/(L/H)8/M/(H/L)8/M/(L/H)3,
- S(2) ≡(H/L)3/M/(L/H)7/M/(H/L)3/M/(L/H)3/M/(H/L)7/ M/(L/H)3,
- S(3) ≡(H/L)2/M/(L/H)6/M/(H/L)6/M/(L/H)2,
- S(4) ≡(H/L)2/M/(L/H)5/M/(H/L)5/M/(L/H)2.
As the first structure, we consider an MPC in the form of S(1)/M/S(1), which we call it . The total thickness of this MPC is 16.59 μm and there are 95 layers in the structure. As can be seen from Fig. 1(a), the transmittance and the Faraday rotation of at ϕ =0 are T=97.09% and θF =75.09°, respectively, and the bandwidth of this structure is about 1.5 nm. By rotating the applied saturation magnetic field and hence, adjusting the magnetization rotation angle of Ce:YIG layers, can reveal the properties of a perfect MOI. This happens at the magnetization rotation angle of ϕ =52.97° for which the off-diagonal elements of the dielectric tensor are ε2xy=5.420×10−3 and ε2xz=7.184×10−3. Figure 1(b) illustrates the operational parameters of this MPC as functions of ϕ and as can be seen, the adjusted shows a negligible ellipticity, a nearly complete transmittance T=99.03% and the Faraday rotation of θF =45°. According to Fig. 1(c), this perfect MOI with acceptable bandwidth of 2 nm provides good performance at 1550 nm.
In order to enhance the transmission bandwidth, we have also carried out a research on contrast between refractive indices of the dielectric layers. In this way, we fixed the high refractive index and varied the refractive index of L layers. By replacing SiO by GGG with refractive index of 1.926 , hence reducing the contrast between dielectric layers, the MO responses will be as T=99.7% and θF =54.05°. We call this structure , whose corresponding adjusted MOI has MO responses as T=99.82% and θF =45°. These responses are achieved at ε2xy=7.515×10−3 and ε2xz=4.951×10−3 which are corresponding to the magnetization rotation of ϕ =33.38°. Figure 2(a) shows the MO properties of this adjustable MOI whose total thickness is 16.20 μm. It can be seen that reducing the contrast between dielectric layers has led to a perfect MOI with larger transmission bandwidth around 2.5 nm. We have also examined another MPC structure constructed upon the basic structure S(1) with GGG as L layer. This MPC, , is in the form of S(1)/M/S(1)/M/S(1) and has the MO responses as T=99.57% and θF =81.12°. By tuning the applied magnetic field we obtain a perfect isolator with T=99.89% and θF =45°. These results are attained for ϕ =56.1° with ε2xy=5.020×10−3 and ε2xz=7.470×10−3. The MOI has the total thickness of 24.48 μm and its ellipticity is close to zero. According to Fig. 2(b), this MOI has 2 nm bandwidth with a high bandwidth flatness that is its superiority over the earlier ones.
By using the basic structure S(2) we can further improve the transmission bandwidth. Figure 3 shows the MO properties of the structure which is in the form of S(2)/M/S(2). At saturation normal magnetic field the transmittance and the Faraday rotation of are T=98.85% and θF =56.68°, respectively, and as can be seen from Fig. 3(a) the bandwidth of this structure is around 2.5 nm. There are 115 layers in the structure with the total thickness of 20.54 μm. By varying the magnetization orientation angle to ϕ =37.34° (equivalent to the off-diagonal elements of the dielectric tensor ε2xy=7.155×10−3 and ε2xz=5.458×10−3), it can be observed from Fig. 3(b) that with an approximately complete transmittance T=99.03% and θF =45°, operates as a perfect MOI. The ellipticity of this MOI is negligible, too. According to Fig. 3(c), the bandwidth of the structure after adjusting to an MOI increases to about 3 nm. However, we can achieve a higher bandwidth using the basic structure S(2) by increasing the repetition number. For example, the structure with about 3 nm bandwidth and T=98.33% and θF =84.83°, after adjustement to a nearly perfect MOI shows an enhanced transmission bandwidth around 3.5 nm (see Fig. 4). Nevertheless, this improvement is acquired at the cost of making the structure thicker.
Utilizing the basic structure S(3), we have introduced another MPC that operates as a perfect MOI with a large transmission bandwidth, a considerable flatness and a negligible ellipticity. This MPC with the total thickness of 32.32 μm is in the form of S(3)/M/S(3)/M/S(3)/M/S(3)/M/S(3), which we call it . As can be seen from Fig. 5(a), at magnetization rotation of ϕ =0 the transmittance and the Faraday rotation of are T=99.16% and θF =53.2°, respectively, and the bandwidth of this structure is 6.4 nm. By rotating the applied magnetic field, hence changing the rotation angle of saturation magnetization vector to ϕ =32.26°, this structure with θF =45° and T=99.03% reveals the properties of a perfect MOI. Figures 5(b) and 5(c) show that as well as the structure under normal saturation field, the ellipticity of the corresponding MOI is negligible and its transmission bandwidth is about 6.4 nm.
Hitherto, we exhibited some MPC structures with sufficiently good transmission bandwidth and the capability of functioning as ideal optical isolators. Although, the proposed MOIs confirm the importance of structural formula for desirable operation, however, the employed materials are also very pivotal as well. To show this feature, we point to the structure in the form of S(4)/M/S(4)/M/S(4). This 19.42 μm-thick MPC has 95 layers in the structure and its L layer has been considered to be SiO2 with refractive index 1.495 . represents a light transmission more than 95% accompanied with a saturation Faraday rotation around 50°. Furthermore, the bandwidth of this structure is 7.1 nm (Fig. 6(a)). Figure 6(b) shows the operational parameters of this MPC as functions of ϕ. As can be seen from Fig. 6(c), by adjusting the rotation angle of magnetization vector to ϕ =25.65° that is equivalent to the off-diagonal elements of the dielectric tensor ε2xy=8.113×10−3 and ε2xz=3.895×10−3, operates as a nearly perfect MOI with θF =45° and an approximately complete transmittance T=96.01%. According to Fig. 6, this adjustment leads to 7.2 nm transmission bandwidth for this MOI with an ellipticity close to zero. The superiority of this MOI rather than the MOI is a larger bandwidth, lower thickness, fewer number of layers and smaller required magnetic field rotation angle for adjustment. This advantage has been reached by incorporating SiO2 in the MPC structure , while the other mentioned materials cannot provide such a performance when incorporated in structure. On the other hand, by reduction of the refractive index contrast of adjacent layers and increase in refractive index of certain layers, we cannot achieve thin MOIs with comparatively acceptable operation.
It should be noted that, the flatness of transmission bandwidth is an important parameter in designing of flat-top MOIs. The transmission spectra can be analyzed by defining a ripple factor and a Faraday rotation flatness parameter that are as follows, respectively:Table 1 shows these parameters for all introduced adjustable MOIs. The results in table 1 indicate that increasing the number of magnetic defects in an MPC structure may increase the transmission bandwidth, but this is not accepted as a rule. Indeed, the deterministic features are the structural formula and the employed materials in the MPC structure.
3.1. The effect of thickness error
For a practical purpose, the influence of the error in thickness of individual layers on the operational parameters of the MOIs should be considered. In the fabrication process of multilayers, the layers are not quite sharp and so, their surface roughness may lead to a deviation in the operation of the devices. Hence, in order to inspect the performance deviation of the introduced MOIs, we consider a set of random thickness errors for the layers. In this technique, surface of each layer is divided into 100 equal parts and a random roughness error is allocated to each one of these portions. This type of error apportioning represents a real form of interface roughness and gives a good estimate for practical results by evaluating the statistical average of a large number of random calculations for the Faraday rotation and the transmittance.
It has been realized that all introduced structures in this paper can provide the operational parameter values of perfect MOIs. Among them, we select the structure to exert the thickness error. In fact, according to the purpose of this manuscript that is the design of flat-top adjustable MOIs, the structure having larger bandwidth is desired. On the other hand, the parameters such as number of defects, total number of layers and total thickness are important for miniaturization. Even more, this MPC requires the smallest magnetic field rotation angle for adjustment. From these regards, we focus just on the MOI structure for error analysis. At first, we consider a random thickness error between −0.2 nm and 0.2 nm for every layer in each portion [18, 46, 47]. In this case, regarding the existence of 95 layers in , the maximum possible difference in the heights of the proposed sections could be 38 nm. For 0.2 nm thickness error, the average values of the Faraday rotation and transmission of the MPC are θF =49.85° and T=93.38%. It can easily be seen that approximately the same results are achieved for a very higher number of segments, hence, this MPC structure can confidently be adjusted to find a Faraday rotation of 45°. By changing the rotation angle of magnetization to ϕ =25.77°, the structure operates as a nearly perfect MOI with θF =45°, a negligible ellipticity and an approximately complete transmittance T=93.97%. According to Fig. 7(a), the transmission bandwidth of this MOI has not been varied by 0.2 nm thickness error of individual layers. This reveals that in addition to making transformation of MPCs to MOIs, the adjustment procedure can effectively compensate the undesirable influence of thickness error on the operation of the MOIs.
It should be indicated that the amount of 0.2 nm thickness error is the least one and is inevitable even if a high-resolution method is employed for fabrication of multilayered MPCs. Therefore, for a more practical purpose, we have also studied the effect of a set of random thickness errors between −1 nm and 1 nm for each part of the layers. In this case, the maximum possible difference in the heights of the proposed portions in the structure could be 190 nm. After applying the mentioned thickness error, the MPC achieves an average Faraday rotation of θF =50.65° and an average transmittance of T=63.27%. By changing the rotation angle of magnetization from ϕ =0° to ϕ =27.26°, the MO parameters are converted to θF =45° and T=63.55%. Thus, the structure is able to be readjusted to an MOI with θF =45° even in the presence of 1 nm thickness error for individual layers, but the amount of the transmission is far from T=100%. However, this amount of transmission is completely acceptable for optical isolators in practice . Figure 7(b) illustrates that the average value of ellipticity is increased and the transmission bandwidth is lessened by growing the roughness. Although, still operates as an acceptable practical MOI, but our error analysis indicates that a high-resolution method is required for constructing the multilayered MOIs as devices operating in certain conditions.
We have investigated the transmission properties of several MPCs to elucidate the possibility of designing adjustable flat-top MOIs with the operating parameters of T≈100% and θF =45°. The adjustment to an MOI is carried out by rotating the applied saturation magnetic field. The established perfect MOIs are thin and contain a low number of magnetic defect layers. The form of these MOIs verifies that the structural formula as well as the materials constructing them is very crucial for achieving flat-top responses with acceptable flatness. On the other hand, a concurrent reduction of the MPC thickness and enhancement of the transmission bandwidth is not attained by increasing the refractive index contrast of the adjacent layers in the structure. We have also considered a thickness error for each layer to evaluate the performance deviation of the introduced MOIs. Our results indicate that for minor error values in thickness, the introduced MOIs have the capability of readjusting to practical MOIs. However, the error analysis proposes a high-resolution fabrication technique for construction of nearly ideal flat-top MOIs.
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