Transfer matrices for one-dimensional (1-D) multi-layered magneto-optical (MO) waveguides are formulated to analytically calculate the nonreciprocal phase shifts (NRPS). The Cauchy contour integration (CCI) method is introduced in detail to calculate the two complex roots of the transcendental equation corresponding to backward and forward waves. By virtue of perturbation theory and the variational principle, we also present the general upper limit of NRPSs in 1-D MO waveguides. These analytical results are applied to compare silicon-on-insulator (SOI) based MO waveguides. First, a three-layered waveguide system with MO medium is briefly examined and discussed to check the validity and efficiency of the above theory. Then we revisited the reported low-index-gap-enhanced NRPSs in MO waveguides and obtained substantially different results. Finally, the potential of common plasmonic waveguides to enhance the nonreciprocal effect is investigated by studying different waveguides composed of Metal, MO medium and dielectrics. Our study shows that the reasonable NRPSs can be optimized to some extent but not as much as claimed in previous publications.
© 2012 OSA
Non-reciprocal photonic devices, including optical isolators and circulators, are always challenging but indispensable in building chip-scale opto-electronic integrated circuit (OEIC). The main obstacles are the large lattice and thermal mismatch between the magneto-optical (MO) garnets and the semiconductor substrates. Considerable efforts have been devoted to break the time-reversal symmetry and produce nonreciprocal effects on the silicon-on-insulator (SOI) platform. A possible solution to bypass the materials problem is to realize one-way mode transition through material index modulation [1, 2], which is appealing because no foreign material is involved beyond complementary-metal-oxide-semiconductor (CMOS) compatible materials. However, further study is required to reduce the device footprint and compensate for the accompanying loss. Hence, conventional structures containing magnetized MO materials are still dominant for producing reciprocal effect. The first successful experiment that brings MO materials to the SOI platform was conducted by direct wafer bonding, which enables the integration of a single-crystal MO film with thin a flat thin SOI waveguide . Very recently, a polycrystalline garnet bilayer containing Ce-doped yttrium iron garnet (Ce:YIG) was directly deposited onto the silicon platform  and an optical isolator was monolithically integrated. This striking technology can be added into the back-end of CMOS process, which enables plenty of room to envision new structures on SOI platform.
The number of theoretical and numerical studies of SOI-based MO waveguides continue to increase [5–9]. It is well known that the Faraday rotation coefficients in widely-used low-loss MO materials (e.g. Ce:YIG) are so weak that the length of the constructed Mach-Zehnder-type isolator is usually at the scale of a millimeter [10, 11]. Therefore, in order to reduce the interaction length of light and MO medium, enhancing MO effect in waveguides has become a hot topic. Different resonators containing MO medium may be feasible since the required nonreciprocal phase shift (NRPS) only needs to shift the backward resonance spectrum away from the forward peaks with a scale of half maximum bandwidth (HMBW) [12,13]. But designs are limited by the operation bandwidth and only work as a nonreciprocal filter. In practice, most isolators are based on interferometers with NRPSs. There are mainly three approaches to enhance the NRPS in waveguides: (1) breaking the waveguide symmetry to cater to the NRPS of the concerned waveguide mode; (2) strongly confining light into a small volume and strengthening its interaction with the MO material and (3) creating additional magnetic domains. However, due to the difficulty in material synthesis, very few research groups have fabricated MO waveguides and most NRPS numerical studies have not been experimentally verified. Hence, analytical calculations are of great importance to study the fundamental physics in MO waveguides and some comparison analysis is useful to improve the reliability.
In this paper, the universal transfer matrix of a multilayered MO waveguide system was derived to accurately calculate the NRPSs, together with the detailed implementation of the uncommon Cauchy contour integration method to find solutions of transcendental equations. The analytical results are in good agreement with those predicted by perturbation theory. We found that there is an upper limit for the available NRPS using a fixed MO material. These theoretical predictions are applied to make comparisons between reported schemes to enhance nonreciprocal phase shifts, such as asymmetric three-layered waveguides, slot waveguides and plasmonic waveguides. By doing so, some ambiguity in previous publications about the calculation of NRPS are cleared up.
2. Transfer matrices for multi-layered magneto-optical waveguide system
Let us start from the time-dependent Maxwell equations to derive the transfer matrix of a homogenous MO layer. The configuration is shown in Fig.1. For the sake of convenience, we use the dimensionless notations in the whole text by normalizing a unit physical length (e.g. 1μm) to unity  and assume the time and z dependent phase factor to be exp [j2πν(t – γz)], where ν is the normalized frequency and γ is the normalized propagation constant (also known as the effective refractive index). The latter can be negative for backward propagating waves. The Faraday’s law and Ampere’s law can be written as the following set of equationsEq.(3) into Eq.(2) yields Eq.(6) into Eq.(4), we get the differential equation for Hy component as 15], indicating that in a homogenous MO medium without waveguiding, the properties of Hy are conserved and no NRPS exists except for a minor modification to the propagation constant. The NRPS must be attributed to the coupling of the two electric fields at the MO material interfaces. By defining and recalling the momentum conservation, we can get the normalized transverse wave vector κ Eq.(7), the solutions of the magnetic field Hy can be separated to a forward wave and a backward wave and take the form of Eq.(8) assures that the An term is for the forward wave with phase retardation along x direction. Subsequently, by using ∂z = − j2πνγ and substituting Eq.(9) into Eq.(6), the field Ez can be written as Eq.(11), we are able to extract the transfer matrix Sn of the n-th layer, which relates the weighted indices An and Bn with An+1 and Bn+1 by
3. Cauchy contour integration method
In later sections, we will study plasmonics-enhanced MO waveguides, in which material loss must be taken into account. Hence, the roots for propagation constants are complex. It is widely accepted that finding the roots of a transcendental equation in a complex plane is quite challenging. Researchers in plasmonics field usually utilize pure numerical root finding schemes, such as Nelder-Mead miniaturization method [16, 17] and reflection pole method . Here we introduce a semi-analytical method to calculate the modes in complex plane by Cauchy contour integration (CCI) [19, 20]. We describe the method as semi-analytical because the root is obtained by an analytical formula of CCI but the integration itself would be done through numerical accumulations.
Originally, the CCI method was frequently used to calculate the residues of a complex function in complex analysis. The basic concept can be described as follow. In a complex plane, there is a domain D enclosed by a simple closed curve C. A single-valued function f(z) is analytical in D and has no null point on C. The count and the sum of roots of equation f(z) = 0 in domain D can be calculated by19], we propose an intuitive and easy-to-implement numerical method to search for the roots. (1) In the complex plane we can plot the figure of numerically and define a trianglar C through P0 – P1 – P2 – P0 to enclose the domain D that contains the poles. (2) N is calculated by Eq.(18) to check the number of roots. Let us choose a reasonable numerical tolerance ε (in our case, ε = 10−4.) (3) If |N| < ε, stop here and recheck the predefined domain C since no root exists in it. (4) If |N − 1| < ε, return the root value evaluated by Eq.(19), else divide the triangle domain C into three child triangles as P0 − P1 − P3 − P0, P1 − P2 − P3 − P1 and P2 − P0 − P3 − P2. (5) For each child domain, go to step (2) for further calculation. The integration should be conducted clockwise otherwise the sign of the result would be reversed. Triangular domains are recommended because they approach the root faster than other polygons and maintain a similar shape between father and child domains.
As needed in Eq.(19), the transfer matrix of the derivative of ST,22 with respect to γ can also be easily derived by a simple linear algebraic manipulation from Eq.(14). All numerical integrations can be easily performed by built-in functions in Matlab. To obtain a continuous and smooth characteristic curve by parameter scanning, for example, the mode dispersion curves, a useful trick is that the triangular domain for the third root can be estimated from the two previous roots.
4. The upper limit of NRPS in 1-D waveguides
Perturbation theory is widely used to calculate the NRPS, especially for the irregular two dimensional (2-D) MO waveguides . Usually the gyrotropy is two orders of magnitude smaller than the reciprocal permittivity, and thus the non-diagonal elements Δεr in the permittivity tensor can be treated as a first-order approximation. Δγ can be estimated by the expression [6, 22, 23]Eq.(20) to cater to the numerical simulation as 23]. Nevertheless, Eq.(20) can also be reduced to another simplified form with the principal magnetic field Hy as  Eq.(22) and Eq.(32).
Since the power carried by a waveguide must be finite, the field components and their derivatives should have maximum values in the waveguide cross section. We expect that there is an upper limit for the absolute value of Δγ. By removing the ∫ dy in Eq.(32), the formula reduces to the one-dimensional case and we have
By neglecting the refractive index discontinuity and using a scalar approximation, the variational principle for eigenvalues show that there is stationary expression for γ2 with respect to small variations of the eigen-field . For the transverse electric modes, we have the following relation
5. Magneto-optical waveguides on SOI platform
In this section, we calculate the NRPS in many different types of MO waveguides on SOI platform to verify the above theory. Unless otherwise specified, the refractive index of Si, Si3N4, SiO2 and Ce:YIG are assumed to be 3.477, 2.0, 1.444 and 2.22, respectively. The values of Faraday rotation of Ce:YIG films was reported experimentally to be as high as −4500°cm−1 in the wavelength of interest, which corresponds to a non-diagonal element of 0.0086 in the permittivity tensor. For the comparison analysis with prior publications, we conservatively assume εxz = 0.005 in the whole text. Actually, all the calculated NRPSs can be scaled to εxz for any realistic MO material.
The non-uniform Yee-meshed full vectorial mode solver was used to calculate the field distributions of all the six mode components, which were substituted into Eq.(20) for NRPS calculation. A sufficiently large computation window surrounded by perfect electric conductor (PEC) boundary conditions were used to enable us to roughly distinguish between leaky wave and guided modes because the former may hit the PEC boundary. In order to reflect the influences of small waveguide dimensions, the minimum mesh size is chosen to be a fraction of the minimum linewidth. As we will see, the results are in good agreement with several individual publications.
5.1. Multi-layered MO waveguides
The three-layered structure is the earliest reported MO waveguide and is widely used in literatures and experiments. Here we calculated the NRPS of the dominant (lowest) modes in such waveguides that can be realized on the SOI platform. Basically, the MO medium can be placed either in the middle or top layers. In light of , the transcendental equation can be deduced from Eq.(14) and expressed asEq. (28) as 15] or sided layer . By taking b01 and b21 as small quantities, one can relate Δγ to γ0 by Eq. (29).
In Fig. 2 we plotted the dependence of NRPS on the waveguide core layer thickness. It can clearly be seen that there is an optimal thickness for the largest NRPS in each structure, whose value is influenced by the refractive index contrast and the extent of waveguide symmetry. Among these structures, those with symmetric refractive index and asymmetrical MO element distributions, i.e. MO(+)/Si/MO(−) and SiO2/MO(+)/MO(−)/SiO2, have the largest NRPSs due to the constructive contributions from both sides of the waveguide. However, such waveguides are more difficult to realize in practice due to the opposite magnetization required. As for the uni-magnetized structures, the MO/Si/Air structure has the highest NRPS due to the serious geometric asymmetry. The optimal thickness gradually decreases to zero in the process approaching the symmetric case. For each curve, the NRPS value decreases rapidly below the optimal thickness to zero since there exists a cutoff thickness for an asymmetric waveguide when γ is equal to the larger refractive index n2 of the two claddings, which can be written as
Figure 2(b) provides us with similar information in the case of center-placed MO medium. The optimal thicknesses are greater than those in Fig. 2(a) due to weaker light confinement. The NRPS is only a small fraction of those in Fig. 2(a). This phenomena is due to the cancelation of MO effect at the two interfaces. Another more intuitive evidence for this explanation is that no NRPS exists in both the MO/Si/MO and SiO2/MO/SiO2 structures because of the cancelation. For a symmetric three-layered structure, we have and b01 = −a01b12 to simplify Eq. (28) so that all the linear terms for γ are missing in the nominator and denominator of Eq. (28). As a consequence, NRPS vanishes.
According to Eq.(27), the largest NRPS one can obtain from any waveguide composed of MO medium, Si and a low-refractive-index material should be roughly smaller than . It should be noted that all the calculated NRPSs fall into this range. In Fig. 2(b), |Δγ| <13.7rad/mm for the SiO2/MO/Air structure. These results agree well with many reasonable results reported in previous publications [3, 10, 11, 15].
5.2. Gap-assisted MO waveguides
In recent years, low-refractive-index (LRI) slot have attracted much attention because of its ability to strongly confine light within a small volume. Therefore, the light interaction with materials can be greatly enhanced . A nano-gap structure was claimed to be useful for enhancing the NRPSs to greater than 30rad/mm (around 52rad/mm for εxz = 0.0086) in a 2-D Ce:YIG/Gap/Si/SiO2 waveguide. If this effect is efficient, the air gap can be readily replaced by another LRI material that can be controlled very accurately in realistic fabrication, e.g. SiO2 film, to enhance the MO effect. However, our results show that NO enhancement can be made by such a sided slot.
Figure 3(a) shows the analytical results for the Ce:YIG/Air/Si/SiO2 structure, which is a four-layered problem. Our method avoids the integration of a discontinuous function in perturbation theory and excludes uncertainty in 2-D cross-section simulation arising from integrals being taken at infinitesimal distances from lines of discontinuity [7, 22]. The thickness of the high-index layer (Si layer) and the gap width were scanned from 0.1μm to 1μm, and from 1nm to 100nm, respectively. We can see that the NRPS decreases monotonically with increasing gap width, without any finite gap width specially good for enhancement.
In order to resolve this problem, the mode profiles for different gap widths d are shown in Fig. 3(b). For step-like dielectric waveguides, the double integral in Eq. (32) is converted into a sum of single integrals along material boundaries xh by 
5.3. Slotted waveguide with a compensation wall
It was also reported that the NRPS can be greatly enhanced by introducing a compensation wall into a slotted waveguide, where the MO material with opposite magnetized direction are sandwiched by high-index dielectric (e.g. silicon). By doing so, 20 times NRPSs enhancement increment and up to 100rad/mm were reported, which seriously exceeds our prediction for the upper limit. After revisiting this problem, we present totally different results.
Firstly, for comparison, the NRPSs in a 1-D Air/Si/MO(+)/MO(−)/Si/Air waveguide resembling the 2-D case is analytical calculated, where ”+” and ”−” denote that the magnetization direction is along +y and −y direction, respectively. As shown in Fig. 4, for x = 0, the NRPS is zero with no MO material. Increasing the MO layer thickness will increase the the NRPS to a maximum value of 6rad/mm. The NRPSs will then decrease and stablize at a value of 5rad/mm for increasing the MO layer thickness. This tendency can be easily explained by the separation of the two silicon layers. If they are far away from each other, the slot effect is missing and the waveguide system is actually two isolated asymmetric waveguides, as shown in Fig. 5(a). The dotted line in Fig. 4 is for the realistic 2-D silicon/MO(+)/MO(−)/silicon waveguide with air cladding and oxide substrate. The NRPS monotonically increases with the increasing slot width instead of any enhancement due to slot effect and finally tend towards the same limit of the 1-D case due to separation of the two silicon rectangular waveguides, as shown in Fig 5(b–c). This result is totally opposite to that in . The reason why there is no enhancement is because of the cancelation of NRPS at the central MO(+)/MO(−) wall (x = 0) and the two sided MO/Si walls ( ) of the slot. Thus, it is impossible to obtain a 20 times enhancement by introducing a compensation wall. Next, we also examined the MO rib waveguide with a compensation wall, which was investigated in  and borrowed as a benchmark in . The varying of NRPS are also shown in Fig. 4 and Fig. 5(d), which are in very good agreement with the results in . The relatively small NRPSs should be attributed to the low index contrast of the calculated model, where the guiding material is a doped YIG with a refractive index of 2.2 and the substrate GGG has an index of 1.9. The mode area in Fig.5(d) is much larger than those in Fig.5(b–c). So, we do not think it is reasonable to compare a large-index-contrast YIG/Si slotted waveguide with a low-index-contrast YIG/GGG rib waveguide.
5.4. NRPS in plasmonic-enhanced MO waveguides
As an important branch of nanophotonics, plasmonics have been developed rapidly for nearly one decade for their potentials in light routing and manipulation . By breaking the diffraction limit to squeeze light into a subwavelength volume, weak physical effects in optical materials can be dramatically enhanced so that the light-matter interaction lengths or volumes can be greatly reduced. Not surprisingly, how the surface plasmon wave propagates in a plasmonic MO waveguide is an interesting topic. Researchers have investigated optical isolating action in surface plasmon polaritons , where large NRPSs can be obtained by eliminating the unwanted cancelation caused by waveguide symmetry. As an extension, here we study several types of mainstream 1-D plasmonic waveguides to discuss their nonreciprocal properties, including the cases of (1) Cu/Si/MO, (2) Cu/MO/Si, (3) Cu/Oxide/MO, (4) Cu/MO/Oxide, (5) Cu/MO(+)/MO(−)/Cu and (6–7) MO(+)/Cu/MO(−). The transverse electric field distributions of the modes of interest are all shown in Fig.6(c). For the MO(+)/Cu/MO(−) structure, there are two fundamental modes: (6) long-range plasmon polariton with higher γ and (7) short-range plasmon polariton with lower γ. For the reason of symmetry, a compensation wall is set for the case (6–7) to obtain the largest NRPS.
The CMOS-compatible metal for supporting plasmonic wave is assumed to be copper, whose material index is modeled according to experimental data . In practice, an NRPS of π is usually necessary to be introduced into one Mach-Zehnder interferometric arm for complete isolation. The required physical length is defined as Lπ (for the case of push-pull scheme, ). Similar to , we also use 1-dB attenuation length L1dB ofFig.6(a–b) by increasing the critical parameter of waveguide thickness. In general, more or less, there is finite accompanying non-reciprocal loss in plasmonic MO waveguides, but it is negligible compared with the reciprocal loss.
Let us read Fig. 6(a) first from the aspects of cutoff point, optimal thickness and limit. First, the cases (1–4) belong to asymmetric three-layer waveguides with metal cladding, so all of them have a cutoff thickness . The cutoff thickness of the cases (2,3) are around 20nm due to quasi-slot effect of LRI guiding, whereas that of the cases (1,4) are greater than 100nm (not shown in figure). Moreover, because NRPS is seriously influenced by the extent of asymmetry, there are optimal thicknesses for NRPS in the cases (1,2). Second, when d = 0 for the cases (2,4), the MO layer vanishes; Lπ goes to infinity and nonreciprocal effect is missing. Conversely, when d = 0 for the cases (1,3) and d → ∞ for the cases (5–7), all the structures return to the simplest bilayer surface plasmon waveguide Cu/MO discussed in , where the transcendental problem of Eq. (17) reduces toEq.(32). For the short-range surface plasmon polariton in the MO(+)/Cu/MO(−) waveguide, the NRPSs also rapidly increase with decreasing layer thickness due to the enhanced fields at . Third, more importantly, we need to compare the magnitudes of Lπ and L1dB, which is critical to the feasibility of combining plasmonics and MO medium. The data in Fig.6 (a) and (b) are plotted differently in magnitude by 3 order. As calculated for the bilayer Cu/MO case, an insertion loss at the scale of 100dB is almost meaningless for practical use in that even a dielectric waveguide can achieve the comparable Lπ without any metallic loss. Just considering enhancement, the case (7) may be appealing because an Lπ value of 100μm can be obtained when d ≈ 10nm, which is very challenging for dielectric MO waveguide. However, the loss problem is very serious due to the relatively fast decrease of L1dB. To conclude, the loss problem seriously constrains the use of plasmonics for enhancement of NRPS.
In conclusion, we analytically investigated the general properties of complex MO waveguides by establishing the transfer matrix of multilayered MO waveguide systems. The enhancement of NRPSs in asymmetric three-layered waveguides, nano-slot waveguides and plasmonic waveguides are revisited and comparisons between the schemes were made. These reported schemes are indeed useful for improving the MO effect, but from an optical analysis, it seems impossible to enhance the MO effect by an order of magnitude. How to bypass this criterion is left as an open issue.
The author H. F. Zhou would like to thank Prof. Kazuaki Sakoda of National Institute for Materials Science (NIMS) in Japan for his enlightening discussion. This study was supported by Singapore SERC/A*STAR Grant 092-154-0098.
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