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We analyze an on-chip optical isolator based on direction dependent single-mode cutoff, which is described in 1D and 2D momentum space. Isolation is shown using 3D finite difference time domain (FDTD) where the magnetization is represented by imaginary off-diagonal permittivity tensor elements. The isolator designs are optimized using perturbation theory, which successfully predicts increased isolation for rib waveguides and structures with non-magnetic dielectric layers. Our isolators are based on bismuth iron garnet and its compatible substrates; an isolation ratio of 10.7 dB/mm is achieved for TM modes.
©2009 Optical Society of America
1. Introduction
Since an isolator is a basic component of information networks, it is important to develop waveguide isolators that can be integrated with the new generation of nanoscale optical circuitry. Compared with other basic components of sub-wavelength, on-chip optical circuits—such as light sources, resonators, and waveguides—the waveguide isolator has not been as deeply studied.
The predecessor to the on-chip isolator is the Faraday rotator-type isolator [1]. Several aspects of the Faraday rotator have been adopted for on-chip isolators—including the materials and some basic theories of magnetooptics. A commonly used material for both rotators and integrated isolators is bismuth-substituted yttrium iron garnet BixY3-xFe5O12 [2]. Also, the fundamental conditions for nonreciprocity apply to both rotator and integrated isolators: the structures must break the time reversal T and spatial inversion S symmetries. Magnetic material can be used to break T and S in various ways: a 1D photonic crystal with periodically layered magnetic and dielectric materials, or an asymmetrical distribution of magnetic material in the plane of the waveguide’s cross-section [3,4]. Alternatively, the conditions for breaking T and S have been analyzed using an analogy between the one-way, nonreciprocal modes generated at interfaces between magnetic and non-magnetic materials and edge states in integer quantum hall systems [5]. Faraday rotators can provide 20 to 30 dB of isolation with only 1–2 dB insertion loss, but the need for polarizers renders this design too bulky for on-chip applications.
On-chip isolators utilize magneto-optical effects other than Faraday rotation, such as a nonreciprocal propagation constant and the nonreciprocal loss phenomenon. Isolators based on dual-waveguide, Mach-Zehnder interferometers (MZIs) operate based on a nonreciprocal propagation constant [6]. Another isolator structure by Amemiya et al. is an active InP-based waveguide isolator operating via nonreciprocal absorption loss for the TM mode with a MnAs magnetic layer and a MQW guiding region [7]. In an FDTD simulation, Amemiya’s isolator achieved 6 dB/mm isolation with -27 dB/mm absorption loss; their fabricated isolator reached 7.2 dB/mm. The largest reported isolation for an integrated waveguide was an InP-based structure with 14.7 dB/mm isolation for TE modes [8].
2. Isolator design and nonreciprocal single-mode cutoff
The main difference between our designs and past integrated isolators is that ours operates via nonreciprocal single-mode cutoff. Nonreciprocal cutoff has been explored in previous studies [9,10], but these reports only analyze 2D waveguides. The single-mode cutoff conditions for a 3D waveguide can be analyzed using momentum space analysis.
Bismuth iron garnet (BIG) is regarded as a promising material for optical isolators due to its good transmission properties and strong magnetooptical response for blue to near-IR light [2]. Specifically, there are strong diamagnetic transitions for BIG at 2.6 and 3.15 eV; these peaks have a tail that extends into the 1 and 2 eV light range, lending the material to applications in telecommunications [11]. Nonzero off-diagonal permittivity tensor elements are characteristic of magnetic materials and they quantify a material’s magneto-optical response. For this paper, we define ε 0 = ε 0′+i ε 0″ as the diagonal permittivity elements and we use ε 1 = ε 1′+i ε 1″ for the off-diagonal elements; for our coordinate system we choose εyz and εzy as the nonzero off-diagonal components and the positive z-axis as the direction of propagation. For isolators based on a nonreciprocal absorption loss or propagation constant, the value of ε 1″ is an important parameter. The value of ε 1″ can be found from magnetooptical ellipsometry experiments that determine the Faraday rotation and Kerr ellipticity. For instance, MnAs was found to have ε 1″ = 0.27 at 1.55 μm [12], while BIG has ε 1″ = 0.09 in the 3 eV range [2]. MnAs is reported to have a larger ε 1″ than BIG, but it is lossy and requires another dielectric material to create a guiding region. By contrast, BIG is an optically transparent and high index (nBIG = 2.5) material that can be combined with gadolinium gallium garnet (GGG) substrates (nGGG = 1.97) to create sub-wavelength waveguides. Also, yttrium iron garnet (YIG), which has ε 1″ = 0.06 and nYIG = 2.3, can be used in conjunction with BIG and GGG [2,13].
Fig 1. Basic BIG isolator design: (a) Normalized band diagram results for different computational techniques; the GGG light line indicates single-mode cutoff. (b) The B:G structure with its dimensions and coordinate system; light propagates along the z-axis.
Fig. 2. Ey TM mode profiles for different values of a for the un-magnetized B:G structure. (a) The mode is largely cutoff or unguided, providing lower than -150 dB/mm transmission; (b) an intermediate cutoff stage, with -30 dB/mm transmission; and (c) the mode is largely guided, with -3 dB/mm transmission. The black outline defines the BIG guiding region.
The first step in our design is to find the width and height of the BIG guiding region that gives single-mode cutoff for a chosen frequency. Figure 1(a) gives band diagram results for the simple BIG on GGG structure shown in Fig. 1(b), which we call the B:G structure. In the band diagram, the frequency ω is normalized by a′/(2πc) and the propagation constant β is normalized by a′/(2π), where a′ is a reference length and c is the speed of light; the width and height of the BIG guiding region both equal a, which is normalized by a′. The slope of the GGG light line is given by the inverse of the refractive index: 1/nGGG. Modes above this light line are unguided, while modes below the line are guided, and the onset of cutoff is given by the intersection between the mode and the light line. Both a frequency domain eigenstate solver (blue line) and beam propagation method (BPM) software (red line) indicate that cutoff is near ω = 0.34, which determines a′. Using the BPM solver we can find the electric field distribution for the B:G structure, as shown in Fig. 2 for different values of a and the same wavelength. These mode profiles illustrate that a = 1.0a′ is near the onset of cutoff; the degree of cutoff is indicated by how much of the mode has been pushed out of the guiding region and into the substrate, since the energy in the substrate is unguided and lost during propagation.
The next step is to verify that the modal cutoff for forward propagating light is different from backward propagating light for our magnetized isolator in 3D finite difference time domain (FDTD) simulations. For isolation using the coordinate system defined in Fig. 1(b), we choose ε 1″ = εyz = εzy * for TM modes; the other off-diagonal components are zero. To visualize nonreciprocal cutoff we utilize 2D momentum space diagrams, following K. Srinivasan and O. Painter [14], which indicate the amount of vertical radiation loss (the amount of radiation lost to the substrate). The energy-momentum dispersion relationship for a slab waveguide is: (nω/c)2 = kx 2+ky 2+kz 2, where n is the refractive index of the substrate and ky is the vertical momentum component relative to the plane of the substrate. For a waveguide on a GGG substrate, the relation (nGGG ω/c)2 = kx 2+kz 2 defines the “light cone” of the waveguide in (kx,kz,ω) space; modes with small in-plane momentum components (kx and kz) will be unguided and lie within the light cone, as shown in Fig. 3(a). To generate the momentum space distribution we take the 2D Fourier transform of Ey in the xz-plane of the substrate.
Fig. 3. (a) The light cone of a slab waveguide; the dotted line represents cutoff. (b)-(d) 2D Momentum space diagrams for the Fourier transform of Ey measured on an xz-plane in the GGG substrate. The data is from 3D FDTD and corresponds to the first quadrant of a plane in the light cone defined by ω = ω 0. Note that the large isolation ratio (147 dB/mm) is because the value of ε 1″ for BIG is increased by a factor of ten. The total amount of energy in momentum space is lowest for (d) because more radiation is lost to the PML.
We increase ε 1″ by a factor of ten to highlight the change to the 2D momentum space mode distribution for forward versus backward propagating light. For B:G, forward light propagates with -3 dB/mm loss, backward light has -150 dB/mm and the nonmagnetic structure has -30 dB/mm. Note that keeping the light source stationary and switching the magnetization from +x to −x is the same as fixing the magnetization along +x and switching between forward and backward light. Also, altering the magnetization vector has the same effect as changing β: both changing the magnetization from +x to −x and decreasing β increase cutoff. A related idea is the scalability of Maxwell’s equations: decreasing the frequency of the light source has the same effect as reducing a. A comparison of Fig. 2 and Fig. 3 reveals the interdependency of these variables. The nonmagnetic structure in Fig. 2(b) and Fig. 3(c) has -30 dB/mm for a = 1.0a′; decreasing the propagation loss to -3 dB/mm can be achieved by either making a = 1.12a′ as in Fig. 2(c), or setting ε 1″ = 0.9 as in Fig. 3(b).
Fig. 4. (a) The imaginary part of the electric field overlap for the B:G structure. (b) Result from perturbation theory for the B:G structure. The plot indicates that there is cancellation for Iyz because the components at the top and bottom of the guiding region have the opposite sign.
3. 3D FDTD isolation results and optimization using perturbation theory
We measured the isolation ratio of the B:G structure and optimized the design via perturbation theory and 3D FDTD. Since the isolation ratio increases as the propagation loss for the forward and backward waves increases, we normalize by adjusting a so that the forward propagation loss due to cutoff is -27 dB/mm, allowing for comparison with Amemyia’s structure that achieved a 6 dB/mm isolation ratio for TM modes with the same forward loss.
In order analyze how the mode generates isolation we use perturbation theory to find the change to the propagation constant β for forward versus backward light. This involves modeling the off-diagonal permittivity tensor elements ε 1″ as a perturbation to the diagonal permittivity elements and defining each mode as an eigenstate. The derivation is presented in full by L. Tang et al. [15]; the perturbation to β is given by:
where
Here, the off-diagonal elements (x, y) εji are only non-zero in the magnetic materials BIG and YIG. Therefore, visualizing the distribution of Iij can indicate the best distribution of magnetic material to optimize the isolation ratio. Figure 4(a) shows Im[Ey *(x,y) Ez(x,y) for B:G, and Fig. 4(b) gives εzy(x,y) Im [Ey *(x,y) Ez(x,y)]. These plots indicate that negative and positive regions of Iyz cancel each other, thereby reducing the total isolation. This result is consistent with our expectations about isolation when considering the waveguide’s spatial inversion symmetry S: since the mode in the B:G waveguide is fairly symmetric, as the substrate causes only slight asymmetry, a uniform distribution of magnetic material does not significantly break S and foster isolation. Rather, a more asymmetric distribution of magnetic material will further break S and increase isolation. The plot of Iyz in Fig. 4(b) indicates the optimal distribution of εzy(x, y): magnetic material should only occupy the top or bottom of the waveguide, avoiding cancellation as Iyz is integrated over the cross sectional area.
Next we simulated structures with BIG only occupying the top or bottom of the guiding region. There are two ways to accomplish these structures: a rib waveguide and using nonmagnetic dielectric materials in combination with BIG to form a guiding region. To maintain the shape of the mode from the B:G structure, the dielectric material should have the same refractive index as BIG. Therefore, a promising candidate is titanium dioxide in the anatase crystal formation, which has n = 2.5 to 2.9. Figures 5(a)–(c) show these new isolator structures, and indeed the isolation ratio is higher than B:G, reaching 10.7 dB/mm for (c).
Fig. 5. (a) Rib waveguide with 7.3 dB/mm isolation ratio. There is still cancellation in the integral Iyz due to the positive contribution near the BIG-GGG interface. (b) Incorporating dielectric materials (TiO2) with BIG; the isolation ratio is 5.41 dB/mm for this T:B:G structure. (c) The B:T:G structure, which has the highest isolation ratio of 10.7 dB/mm.
The results for all of the structures that we simulated are provided in Table 1. The rib waveguide structure in Fig. 5(a) is referred to as “Rib B:G,” and the other structures are named according to their material layers; for example, Y:B:G refers to YIG layered on top of BIG with a GGG substrate, and T denotes TiO2. The columns ‘Iyz negative’ and ‘Iyz positive’ indicate the blue and red zones in Fig. 5, respectively, while ‘Iyz total’ gives the integral over the entire cross section. Table 1 indicates that B:T:G provides the most isolation, but this structure would require the B and T layers to be grown separately and wafer bonded together. Rather, the Rib B:G structure could be fabricated by growing BIG on GGG and etching the waveguide with a mask.
Table 1. 3D FDTD Isolation data for the various waveguide isolator structures.
4. Conclusion
We demonstrate 10.7 dB/mm of isolation for TM modes in a waveguide based on bismuth iron garnet using 3D FDTD. Isolation is generated using nonreciprocal single-mode cutoff, which is a result of a nonreciprocal propagation constant. The conditions for cutoff are analyzed using both 1D and 2D momentum space mode profiles.
Acknowledgments
The authors acknowledge Innovation Core SEI, Inc. for financial support, and thank you to Akihiro Moto and Lingling Tang for discussions.
References and links
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