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We have implemented the nonreciprocal propagation capabilities into plasmonic waveguides and have simulated the performances. We employed dielectric-loaded surface plasmon polariton waveguide (DLSPPW) and long-range DLSPPW (LR-DLSPPW) configurations, where ferromagnetic-metal Fe is used instead of noble metals in order to obtain nonreciprocal propagations by the transverse magneto-optical (MO) effect. The nonreciprocal performances were characterized by the finite-difference frequency-domain (FDFD) method in terms of the propagation losses in return for the nonreciprocal phase shift (NRPS) and nonreciprocal propagation loss (NRL). The NRPS and NRL of the DLSPPW configuration are larger than those of the previously reported semiconductor waveguide optical isolators owing to the large MO constant of Fe and the field confinement by surface plasmons although the propagation loss for NRL of 1 dB is at least 31 dB and the propagation length is limited to less than 10 μm. To reduce such a large propagation loss, we introduced the LR-DLSPPW configuration composed of Polymethyl methacrylate (PMMA) ridge and Benzocyclobutene (BCB) buffer layer. The Fe layer thickness and width are optimized to 50 nm and 500 nm, respectively, so that sizable MO effect and low propagation loss coexist. The propagation loss for NRL of 1 dB is suppressed to ~10 dB within a waveguide length of ~56 μm. Our comprehensive investigation offers fundamental information on practical magneto-plasmonic waveguides and how much nonreciprocal performances are expected, providing an insight into the integration of magneto-plasmonics with on-chip photonics and electronics.
© 2017 Optical Society of America
1. Introduction
Surface Plasmon Polaritons (SPPs) are quasiparticles arising from the quantization of collective electron oscillations coupled to light waves at the interface between a metal and a dielectric. The concept of SPPs was first regarded over decades ago, and since then the technologies associated with SPPs, plasmonics, have been widely developed. Plasmonics is expected to play an important role in chip-scale electronics and photonics [1]. The ever-increasing demand for faster information transport and processing speed is calling for both high-dense miniaturization of electronics and high-speed communication of photonics. The SPP potentially possesses both of them, because it has the nature of light wave with subwavelength confinement beyond the diffraction limit. To mature plasmonic/photonic integrated circuits (PICs) based on this demand, not only passive elements but also active interaction with SPPs is necessary. Up to now, a lot of achievements, e.g. SPP-enhanced light emitting diodes or laser diodes [2–4] and modulators of SPP propagations [5,6], have been made.
One of the ways for the active interaction with SPPs is using magneto-optical (MO) effect. It has been reported that the propagation constant and losses of SPPs can be modified by the MO effect [7–11]. The unique feature that the MO effect offers is breakage of the time-reversal symmetry. Usually, homogeneous media of symmetric permittivity and permeability tensors must obey Lorentz reciprocity theorem, and thus light in those media does not differ for the reversal of the propagation direction. There are only a few cases that enable nonreciprocal propagation of the light; namely, the permittivity and permeability are time-dependent, nonlinear materials, or the MO materials [12]. Therefore, incorporating the MO effect with SPPs (MO-SPPs) would bring about the on-chip miniaturization along with the nonreciprocity and is expected to realize optical isolators for PICs and moreover optical diodes for all-optical sequential logic circuits.
There have been several proposals for unidirectional propagation of MO-SPPs for PICs [8–11,13,14]. Unfortunately, most of their designs were based on one-dimensional optical confinement with configurations difficult to achieve. For example, an MO-SPP waveguide was composed of a noble metal, which is treated as an MO material under very strong external magnetic field [14]. Also, an MO-SPP waveguide consisted of a very thin noble metal film sandwiched by two magnetic garnets being magnetized in opposite directions [8]. In general, magnetic garnets are incompatible with semiconductor-based PICs due to large mismatching of material properties and are difficult of arbitrary modeling. To avoid the high optical loss of SPPs, some of MO-SPP waveguides were assumed to work under nearly radiative propagation condition [8–11], which is inadequate for minute PICs and in contradiction to SPPs’ highlighted small field confinement. In these situations, there has been no report on experimental demonstrations of the nonreciprocal MO-SPP waveguides. In order to design more practical MO-SPP waveguides, designs based on analysis of two-dimensional optical confinement are required. These analyses provide fundamental information on nonreciprocal MO-SPP waveguides including device configuration compatible with PICs.
In this paper, we investigate MO-SPP waveguides based on a Dielectric-loaded surface plasmon polariton waveguide (DLSPPW) [15,16] and its derivative model, Long-range Dielectric-loaded surface plasmon polariton waveguide (LR-DLSPPW) [17,18]. Among various plasmonic waveguide structures [19–22], the DLSPPW and the LR-DLSPPW feature I. relatively easy fabrication process, II. a rather good tradeoff between the mode confinement and the propagation length, III. good coupling efficiency experimentally demonstrated with photonic waveguides [23,24], and IV. scalability of metals and dielectrics, which allows for easy development into active plasmonic devices by introducing gain, thermo- and electro-optic effects [25–28], and even the MO effect. In what follows, we substitute a ferromagnetic metal Fe for the metal part of the DLSPPW and LR-DLSPPW, and theoretically investigate the propagation characteristics including nonreciprocal effects caused by the MO effect while changing the dimensions and refractive indices. The propagation loss in exchange for the nonreciprocal propagation loss is improved by introducing the LR-DLSPPW configuration. We summarize the characteristics of these MO-SPP waveguides, compared with those of the previous works.
2. Calculation method and definitions
Given that a saturation magnetic field is applied perpendicular to the propagation direction of SPPs and in the film plane (parallel to the x-axis in Fig. 1), the Fe has the relative permittivity tensor with off-diagonal components,
where εr is the relative permittivity without magnetization and εMO represents the MO constant caused by magnetization. One can deduce from Eq. (1) that the y and z components of electric fields entangle with each other inside MO materials and cause modulation on the propagation constant of light. This configuration of the MO effects is called transverse MO Kerr effect (TMOKE). Since TMOKE has no need for polarizers, birefringent crystals, or the TE-TM mode matching [29] in slab waveguide structures to block the counter propagation as compared with Faraday effect, it is suitable for integrated optical isolators by using nonreciprocal phase shift and loss [8–14, 29–32].
Fig. 1 (a) A schematic diagram of the DLSPPW configurations with Fe. (b) Closeup of the computation domain, in which the meshes represent the non-uniform Yee grid cells. Positive ( + M) or negative (–M) magnetization is applied in the direction indicated by the red or blue arrows, respectively.
In order to investigate a plasmonic waveguide under the TMOKE, we apply finite-difference frequency-domain (FDFD) method [33,34]. Although there are few literatures on FDFD method, it is based on the same Yee grid scheme as a popular finite-difference time-domain (FDTD) method and the anisotropy caused by MO effect can be implemented by averaging manipulation [35]. To reduce the approximation errors arising from the staggered nature of the Yee grid and the averaging manipulation for the MO effect, the grid cell size should be fine enough particularly at the metal surface. We therefore used an edge-enhanced non-uniform grid, in which the Yee grid cells are becoming smaller towards the boundaries of segments as shown in Fig. 1(b). We set the cell size to simultaneously satisfy the conditions that the cell size is smaller than one-tenth the wavelength inside the highest refractive index medium; the cell size at the metal interface is smaller than 1 nm; and the number of cells inside a segment is at least 10. The FDFD method translates the computation domain into a square matrix and solves it as an eigenproblem. The effective refractive indices, neff, can be solved as eigenvalues, while every electric and magnetic field distribution as eigenvectors.
Notice that it still prevails in the study of integrated optical isolators to calculate nonreciprocal effects by MO effect based on the field distribution without the MO effect and a perturbation theory [29]. Our FDFD method explicitly solves the nonreciprocal effect and the simulations below are unique from this point of view.
According to the solutions the FDFD method provides upon the magnetization reversal, we define propagation characteristics of the MO-SPP waveguide as follows:
Δneff: A change in the effective refractive index upon magnetization reversal,
NRPS: A nonreciprocal phase shift calculated by the real part of Δneff and the wavenumber in vacuum k0,NRL: A nonreciprocal propagation loss calculated by the imaginary part of Δneff,απ/2 NRPS: A propagation loss (the smaller one between + M and –M) in return for NRPS of π/2,α1-dB NRL: A propagation loss in return for NRL of 1 dB,Lπ/2 NRPS: A necessary length for NRPS of π/2,L1-dB NRL: A necessary length for NRL of 1 dB,We use απ/2 NRPS, α1-dB NRL, Lπ/2 NRPS, and L1-dB NRL as figures of merit of the nonreciprocal waveguide. The minima of απ/2 NRPS and α1-dB NRL indicate the best tradeoff between the nonreciprocal effects and the propagation loss.In the following calculations, the wavelength is assumed to be λ = 1.55 μm, where refractive indices (εr = n2) of materials cited in this paper are Polymethyl methacrylate (PMMA): n = 1.493, Cytop: n = 1.335, Benzocyclobutene (BCB): n = 1.535, Fe: n = 3.62 – 5.56 i, εMO = 3.12 + 1.8 i, Ni: n = 3.38 – 6.82 i, εMO = 0.2 + 0.86 i, and Co: n = 3.56 – 7.15 i, εMO = 1.58 + 2.1 i [18,36–38]. The definitions in this paper comply with a convention of expi(ωt – k0 neff z) for the forward propagation in the z-direction.
3. Devices structure and simulations for nonreciprocal DLSPPW configuration
The DLSPPW basically consists of a dielectric ridge placed on a large plane metal surface, and the confinement of SPPs is secured vertically and laterally by the dielectric ridge of higher refractive index than that of surroundings. The device structure and calculation model are shown in Fig. 1. In order to yield TMOKE, we chose Fe as a metal underlayer. Fe has a negative permittivity in the infrared wavelength region (plasma frequency: 4.23 eV, relaxation time: 1.2 × 10−14 s [39]) and can generate SPPs at λ = 1.55 μm. Also, Fe exhibits the largest TMOKE of MO-SPPs among ferromagnetic metals. For instance, NRL of MO-SPPs at air/Fe, air/Ni, and air/Co interfaces are calculated to be 0.014 dB/μm, 0.0014 dB/μm, and 0.0010 dB/μm, respectively. The dielectric ridge of a refractive index of nr is assumed to be built on a wide plane Fe underlayer magnetized in the x-direction and surrounded by air. The Fe layer is thick enough to ensure the semi-infinite film on the bottom side and thus the substrate is ignored. This is valid if the Fe underlayer is thicker than ~120 nm. Compared with a DLSPPW with Au underlayer, whose thickness of ~80 nm is enough to be sufficiently opaque [23], this implies that Fe is less specular and causes high absorption loss to SPPs.
4. Calculation results of nonreciprocal DLSPPW configuration
4.1 Influence of the dimension and the refractive index of the ridge on the DLSPPW configuration
The propagation loss and field confinement in a DLSPPW structure are strongly dependent on the dielectric ridge [16]. We therefore investigate the influence of the dimensions of the dielectric ridge on the nonreciprocal propagation with Polymethyl methacrylate (PMMA) as a dielectric ridge (nr = 1.493) by default and changing the thickness from T = 20 nm to 1 μm for the ridge widths from W = 0.3 μm to 1 μm in Fig. 2.
Fig. 2 The dependence of the nonreciprocal propagation characteristics on the thickness (T) and width (W) of the ridge in the DLSPPW configuration. (a) The real part of the effective refractive index (red and blue solid curves, left axis) and the real part of Δneff (dashed black curves, right axis) related to the NRPS = k0 Re[Δneff]. (b) The propagation length Lspp (red and blue solid curves, left axis) and the NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose. The white curve inside indicates the intensity along the vertical direction at the horizontal center.
The real part of neff (the left axis in Fig. 2(a)) contains two curves of the positive ( + M, red) and negative (–M, blue) magnetization states almost overlapping with each other. The change in the effective refractive index, Re[Δneff], (dashed black curves, the right axis in Fig. 2(a)) represents the displacement of these curves and is directly proportional to the NRPS. The propagation length, Lspp (the left axis in Fig. 2(b)), at which the intensity of SPPs decays by 1/e, also has two curves representing the positive ( + M, red) and negative (–M, blue) magnetization states. A DLSPPW with Au typically shows the propagation length of over 50 μm around T = 0.7 μm [16,23]; however, the DLSPPW with Fe shows only Lspp ≈3 μm there on account of the high ohmic loss of Fe. The upper insets in Fig. 2 show the field distribution of electric field for TM-like mode in + M state. The field profile hardly changes on the magnetization reversal.
Since the MO effect is proportional to the amount of optical field inside an MO material [40], the NRPS (Re[Δneff]) and NRL increase as the ridge height (T) decreases due to increased field confinement to the Fe surface, while they decrease as the ridge width (W) decreases because the field leaks to air. The NRPS (Re[Δneff]) and NRL reach their peaks around T = 0.3–0.4 μm depending on the width. These peaks correlate with the Lspp minima, where the field confinement comes to the inflection point and further reduction of the ridge height results in re-increase of mode field and the propagation length. The reason for the peaks of NRPS (Re[Δneff]) and NRL is that the field is most compressed to the Fe underlayer, enhancing the MO effect.
It was reported that π/2 NRPS was obtained in ~400-μm long Si waveguide optical isolators (~0.00125 π rad/μm), where a cerium-substituted yttrium-iron-garnet ((YCe)3Fe5O12, Ce:YIG) die was bonded as an upper cladding layer [29,30]. The NRPS of the Fe DLSPPW is ~0.004 π rad/μm (Re[Δneff] ≈0.003) for W x T = 1 x 0.37 μm (Fig. 2(a)), which is 3.2 times larger. It was also reported that NRL of 0.0147 dB/μm was demonstrated in an InGaAsP / InP semiconductor optical amplifying (SOA) isolator, where Fe was deposited as a side cladding layer and optical loss is compensated by current injection [31]. The NRL of the Fe DLSPPW is ~0.08 dB/μm for W x T = 1 x 0.27 μm (Fig. 2(b)), which is 5.4 times larger. These enlargements of the nonreciprocal effect are owing to the much higher MO constant of Fe than that of Ce:YIG (εMO ≈0.0085 i [29]) and higher field confinement into Fe with the aid of SPPs.
Another parameter available to be investigated is the refractive index of the ridge. In Fig. 3, we performed the simulations at W = 1 μm for refractive indices of lower, (nr = 1.35), and higher, (nr = 1.75), than that of the default PMMA (nr = 1.493). These indices are obtainable with some resins, glasses and solids.
Fig. 3 The dependence of the nonreciprocal propagation characteristics on the refractive index of the ridge (nr) in the DLSPPW configuration of W = 1 μm. (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose.
The SPPs’ field confinement is proportional to the refractive index of the ridge as shown in the upper insets of Fig. 3. The higher refractive index thereby leads to the higher neff, NRPS, NRL, and shorter Lspp. The NRPS and NRL of nr = 1.75 are increased about twice from those of PMMA.
In order to figure out the optimal condition, where NRPS and NRL are obtained in exchange for the smallest loss and the shortest length possible, Fig. 4 shows parametric plots of απ/2 NRPS versus Lπ/2 NRPS (Fig. 4(a)) and α1-dB NRL versus L1-dB NRL (Fig. 4(b)) as the figures of merit for all the DLSPPW configurations above. Each trajectory indicated by the arrows represents the process of change as the thickness of ridge changes from T = 20 nm to 1 μm.
Fig. 4 Parametric plots of (a) απ/2 NRPS versus Lπ/2 NRPS and (b) α1-dB NRL versus L1-dB NRL for the different widths (black curves) and refractive indices (red curves) of the ridge. The arrows indicate the direction of each trajectory as the thickness of the ridge changes from T = 0.02 μm to 1 μm.
The waveguide parameters for the optimal condition can be found at the plot area towards the bottom left.
Within the PMMA ridge (black curves), the wider widths generally show better values. W x T = 1 x 0.47 μm (indicated by A in Fig. 4(a)) achieves Lπ/2 NRPS = 135 μm, whereas the απ/2 NRPS reaches 357 dB despite being the smallest loss. The smallest α1-dB NRL = 31.0 dB is shown around W x T = 0.3 x 0.17 μm (indicated by C in Fig. 4(b)), where the guided mode is on the verge of radiative mode, lessening the loss. Almost the same value of α1-dB NRL = 31.4 dB is found at W x T = 1 x 1 μm (indicated by D in Fig. 4(b)). Considering the required length, W x T = 1 x 1 μm is preferable, where L1-dB NRL is 17.6 μm.
Variation in the refractive index more largely influences the nonreciprocal performance (red curves). It is obvious that the enhancement of the TMOKE by the strong field confinement by high refractive index of nr = 1.75 reduces the required lengths. Although this confinement is followed by high propagation loss, απ/2 NRPS for nr = 1.75 can be comparable to that of PMMA at T = 0.47 μm (indicated by B in Fig. 4(a)), where απ/2 NRPS = 360 dB and Lπ/2 NRPS = 94 μm. More remarkably, α1-dB NRL is improved by nr = 1.75 in all of given ridge thickness range (Fig. 4(b)). The reduction in α1-dB NRL is saturated around ~22 dB at T = 1.1 μm (indicated by E in Fig. 4(b)). Further increase in the thickness of the ridge is just accompanied by the extension of the required length.
According to the results above, the ridge of a wider and thicker dimension and a moderately higher refractive index is better for the optimal nonreciprocal performances of the DLSPPW configuration. For single mode operation, however, the ridge dimension is limited to about W x T ≈0.6 x 0.6 μm as the conventional DLSPPW with Au is designed.
5. Devices structure and simulations for nonreciprocal LR-DLSPPW configuration
As described above, the NRPS and NRL of the Fe DLSPPW are large. It must be the key for practical use to reduce the propagation loss. It is likely to come up with a long-range SPP mode configuration [41] for this sake. The reason for loss reduction in the long-range SPP mode, such as insulator-metal-insulator (IMI) and metal-insulator-metal (MIM) configuration, is that an electromagnetic field is symmetrically balanced on either side of the metal similar to a transverse electromagnetic (TEM) wave and thereby the electric field component parallel to the metal surface is disappeared. However, it has been reported that the nonreciprocal effect by the TMOKE is canceled if the waveguide structure is perfectly symmetric, including the IMI [8] and MIM [42] configurations. The balanced electromagnetic field can also be imitated by introducing another dielectric film into an asymmetric IMI structure. The LR-DLSPPW achieves the longer propagation length in this manner. Therefore, utilizing the LR-DLSPPW configuration helps reduce the propagation loss while keeping the nonreciprocal effect given by the TMOKE.
The LR-DLSPPW basically consists of a metal stripe and three dielectrics (ridge, buffer, and substrate) of different refractive indices. Atop the substrate of the lowest index, the metal stripe is sandwiched between the ridge and buffer layers of relatively higher indices, whose thicknesses and indices are optimized so that the optical field of the top and bottom sides of the metal stripe is balanced analogous to a long-range SPP mode.
In order to let the LR-DLSPPW be capable of the nonreciprocal propagation, the designed LR-DLSPPW is composed of Fe sandwiched by the dielectrics ridge of PMMA and the buffer layer of a refractive index of nb on the top of the substrate of Cytop as shown in Fig. 5. The buffer layer is assumed to be BCB (nb = 1.535) by default.
Fig. 5 (a) A schematic diagram of the LR-DLSPPW configurations with Fe of variable dimension. (b) Closeup of the computation domain, in which the meshes represent the non-uniform Yee grid cells. Positive ( + M) or negative (–M) magnetization is applied in the direction indicated by the red or blue arrows, respectively.
6. Calculation results of nonreciprocal LR-DLSPPW configuration
6.1 Influence of the dimension of the Fe layer on the LR-DLSPPW configuration
The thickness of the metal stripe of an LR-DLSPPW of Au is typically 15 nm to reduce ohmic loss and still be deposited with a good quality [17,18]. However, for the sizable MO effect, a fair amount of Fe is required. We first optimize the thickness of the Fe film (TFe) with keeping the width (WFe) infinite in the computation domain and the dimension of the ridge W x T = 1 x 1 μm. Figure 6 shows the cases of TFe = 20, 35, 50 and 70 nm. Note that the decisive factor to balance the mode field and hence reduce the propagation loss of the LR-DLSPPW is the thickness of the buffer layer. Thus, every calculation below is conducted as a function of the thickness of the buffer layer, t.
Fig. 6 The dependence of the nonreciprocal propagation characteristics on the thickness of the Fe layer (TFe) in the LR-DLSPPW configuration (W x T = 1 x 1 μm, WFe = infinity). (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose. The white curve inside indicates the intensity along the vertical direction at the horizontal center.
The proper thickness of the buffer layer t to be a balanced long-range SPP mode can be determined by the maximum propagation lengths, as shown in Fig. 6(b). The thinner metal promotes the coupling of SPPs on the upper and lower surfaces of the metal film as well as the smaller ohmic loss and results in the longer propagation length. Thus, TFe = 20 nm shows the longest propagation length of ~14 μm at t ≈200 nm, which is improved from ~2.4 μm of the DLSPPW configuration with the same dimension of the ridge. As indicated in the upper insets in Fig. 6, the optical fields on above and below the Fe film evolve almost separately as the thickness of the Fe increases to thicker than TFe = 50 nm. Therefore, the SPP mode is not bound by the Fe film and the mode field becomes laterally radiative inside the buffer layer above t = 300 nm with TFe = 70 nm. We thus omitted these cases from each graph in Fig. 6.
From the point of view of the nonreciprocal effect, the thicker Fe film brings larger MO effect, while it entails higher propagation loss. Therefore, Lspp and NRL in Fig. 6(b) show totally opposite trends for the Fe thickness.
It is worth mentioning that NRL is derived from the nonreciprocal change in the optical energy flow inside an MO material having a real part of the off-diagonal permittivities such as ferromagnetic metals. On the other hand, NRPS occurs both on the surface and inside of an MO material. That is why the NRL is proportional to the amount of the Fe and the field confinement to the Fe layer, whereas the NRPS shown in all figures within the LR-DLSPPW configurations are rather random as compared with the DLSPPW configuration, indicating the composite influence of the TMOKE at the top and bottom interfaces of the Fe layer as well as the amount of the Fe.
As will be described later, the optimal thickness of the Fe layer is determined to be TFe = 50 nm. For the long-range SPPs supported by the finite width of a metal film, the mode is created from a coupling of both the edge and corner modes, leading to the evolution of the long-range mode. A decrease in the width and thickness of the metal stripe of the LR-DLSPPW makes the mode field closer to the TEM wave and results in lower propagation loss [18,20].
We perform the same calculations as above with the finite width of the Fe film ranging from WFe = 200 nm to 1 μm while keeping the thickness TFe = 50 nm and the size of the ridge W x T = 1 x 1 μm. The calculation results are shown in Fig. 7, compared with the case of WFe x TFe = infinite x 50 nm.
Fig. 7 The dependence of the nonreciprocal propagation characteristics on the width of the Fe layer (WFe) in the LR-DLSPPW configuration (W x T = 1 x 1 μm, TFe = 50 nm). (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose. The white curve inside indicates the intensity along the vertical direction at the horizontal center.
As exhibited in the upper insets in Fig. 7, the optical field is distributed more homogeneously around the ridge than the case of the laterally infinite Fe layer shown in Fig. 6. The propagation length extends inversely proportional to the width of the Fe and reaches Lspp(−M) = 53 μm and Lspp( + M) = 49 μm at t = 300 nm of WFe = 200 nm. The displacement of Lspp( + M) and Lspp(–M) curves for WFe = 200 nm seems wider than the other curves in Fig. 7(b) although the NRL of WFe = 200 nm lies lower. This can be explained that the NRL is proportional to Im[neff( + M)] – Im[neff(−M)], whereas Lspp(−M) – Lspp( + M) ( = 1/2k0Im[neff(–M)] – 1/2k0Im[neff( + M)]) is proportional to (Im[neff( + M)] – Im[neff(−M)]) / (Im[neff( + M)] × Im[neff(−M)]); therefore, the enlarged displacement seen in WFe = 200 nm is accounted for by a reduction in the propagation loss (Im[neff( ± M)]).
The influence of the thickness and width of the Fe layer on the consequential figures of merit is summarized in Fig. 8, in which each trajectory follows the change in the thickness of the buffer layer from t = 40 nm to 500 nm.
Fig. 8 Parametric plots of (a) απ/2 NRPS versus Lπ/2 NRPS and (b) α1-dB NRL versus L1-dB NRL for the different thicknesses with infinite width of the Fe layer (black curves) and different widths with a 50-nm thickness of the Fe layer (red curves). The arrows indicate the direction of each trajectory as the thickness of the buffer layer changes from t = 0.04 μm to 0.5 μm.
For WFe = infinite with different thicknesses (black curves), the smallest απ/2 NRPS of ~480 dB is found at t = 500 nm of TFe = 20 nm (indicated by A in Fig. 8(a)), which is deteriorated from that of the DLSPPW configuration (Fig. 4(a)) because the reduction in NRPS is considerable (compare Re[Δneff] of Figs. 6(a) and 7(a) with Fig. 2(a)). By contrast, the smallest α1-dB NRL is improved to ~20 dB at t = 300–450 nm of TFe = 70 nm and 50 nm (indicated by B in Fig. 8(b)), compared with ~31 dB of the DLSPPW configuration (W x T = 1 x 1 μm indicated by D in Fig. 4(b)). Even though the minimum of α1-dB NRL is slightly better for TFe = 70 nm than for TFe = 50 nm, απ/2 NRPS and available thickness range of the buffer layer is in favor of TFe = 50 nm rather than TFe = 70 nm. Thus, we chose the thickness of the Fe film as TFe = 50 nm.
By using the finite width and TFe = 50 nm (red curves), the propagation loss is substantially suppressed. Despite the smaller amount of Fe than WFe = infinite, the smallest απ/2 NRPS is mitigated to ~450 dB (indicated by C in Fig. 8(a)) from ~780 dB of WFe x TFe = infinite x 50 nm (indicated by D in Fig. 8(a)), owing to the largely improved propagation loss. The smallest α1-dB NRL is also improved to ~10 dB at t = 350–400 nm of WFe = 500 nm and 200 nm (indicated by E and F, respectively, in Fig. 8(b)). The improvement in the figures of merit of the nonreciprocity is almost saturated with WFe = 500 nm. Further reduction to WFe = 200 nm results in a little reduction in the smallest values of α1-dB NRL and απ/2 NRPS, followed by a considerable increase in the necessary length. Therefore, it would be the best to choose the dimension of the Fe layer as WFe x TFe = 500 x 50 nm.
Note that, in an actual device, this Fe stripe layer is long in the propagation direction and short in the magnetization direction of TMOKE; therefore, we should discuss its shape magnetic anisotropy. The easy axis of magnetization in the LR-DLSPPW configuration is in the propagation direction and perpendicular to the magnetization direction of TMOKE. The saturation magnetic field in this case is probably a few kOe [31], which can still be given by commercially available permanent magnets (ferrite magnet: ~3 kOe, neodymium magnet: ~10 kOe).
6.2 Influence of the ridge dimension and the refractive index of the buffer layer on the LR-DLSPPW configuration
In what follows, we investigate other waveguide parameters than the Fe layer, namely the thickness and width of the ridge and the refractive index of the buffer layer. Their influence on the figure of merit for nonreciprocal performance will be summarized in parametric plots later.
Thus far, the dielectric ridge has been fixed to W x T = 1 x 1 μm. The field confinement and consequential propagation characteristics are also influenced by the dimension of the ridge as is the case with the DLSPPW configuration. We analyzed the influence of the width and the thickness of the dielectric ridge with the Fe stripe of WFe x TFe = 500 x 50 nm. First, calculations are performed for different ridge thicknesses of T = 0.7 μm and 1.3 μm, while keeping the ridge width W = 1 μm. Figure 9 shows the calculation results compared with the ridge dimension of W x T = 1 x 1 μm.
Fig. 9 The dependence of the nonreciprocal propagation characteristics on the thickness of the ridge (T) in the LR-DLSPPW configuration (WFe x TFe = 500 x 50 nm, W = 1 μm). (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose.
A reduction in the ridge thickness to T = 0.7 μm interrupts the equilibrium of the mode field on either side of the Fe. The propagation length shows no local maximum and monotonically increases as the buffer layer becomes thinner (Fig. 9(b)) because the mode field leaks to the substrate. The NRPS (Re[Δneff]) and NRL for T = 0.7 μm are larger than those for the thicker ridge until t ≈100 nm (Figs. 9(a) and 9(b)) because the mode field is compressed to the Fe by the smaller ridge. Further reduction in the buffer layer thickness makes the effective refractive index close to the index of the substrate (1.335), meaning that the mode field is largely distributed in the substrate. That is why the NRPS (Re[Δneff]) and NRL decrease rapidly as the buffer layer becomes thinner.
Increasing the ridge thickness to T = 1.3 μm makes the required thickness for the buffer layer to balance the mode field increase as indicated by the Lspp peaks. The propagation length is slightly longer than those of the lower ridge above t ≈250 nm because of slightly weaker confinement to the Fe surface in the higher ridge. That is also followed by a reduction in the NRPS (Re[Δneff]) and NRL as compared with those of the lower ridge.
The propagation properties of LR-DLSPPW are also influenced by lateral confinement controlled by the ridge width W. Fig. 10 shows how the propagation properties are influenced by the width of the dielectric ridge in the range from W = 0.7 μm to 1.3 μm with the fixed ridge thickness of T = 1 μm.
Fig. 10 The dependence of the nonreciprocal propagation characteristics on the width of the ridge (W) in the LR-DLSPPW configuration (WFe x TFe = 500 x 50 nm, T = 1 μm). (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose.
As was the calculation results for the ridge thickness, a reduction in the ridge width to W = 0.7 μm causes a reduction in the virtual refractive index of the ridge, being not enough to balance the mode field on either side of the Fe layer. A width of W = 0.7 μm compresses the mode field to the Fe surface with the thicker buffer layer than t ≈200 nm, enhancing the NRPS (Re[Δneff]) and NRL compared with those for wider ridge widths; however, the curves of the NRPS (Re[Δneff]) and NRL for W = 0.7 μm lies under those for wider widths below t ≈200 nm due to the large radiation to the substrate. As was the case for T = 0.7 μm, this tradeoff limits the best figures of merit of the nonreciprocal performances.
It has been reported that the propagation length and the tolerance for the radiative mode are dependent more on the width than on the thickness of the ridge in the LR-DLSPPW with Au [18]. The same trend is observed with Fe. The monotonic evolution in the propagation length with reducing the thickness of the buffer layer is more rapid for W = 0.7 μm than the case for T = 0.7 μm (Figs. 9(b) and 10(b)) because the light leaks into the substrate more rapidly. This can be also confirmed by the effective refractive index. The curve of Re[neff] for W = 0.7 μm (Fig. 10(a)) lies lower than for T = 0.7 μm (Fig. 9(a)) and reaches the refractive index of the substrate (1.335) around t = 200 nm.
For the wider ridge width of W = 1.3 μm, the thickness of the buffer layer to balance the mode field becomes large similarly to the case of T = 1.3 μm. In contrast to the LR-DLSPPW with Au, in which the ridge wider rather than higher are better for longer propagation lengths [18], the longest propagation length for W x T = 1.3 x 1 μm (Lspp = 26.3 μm (–M) and 24.0 μm ( + M) around t = 400 nm) is comparable to that for W x T = 1 x 1.3 μm (Lspp = 26.2 μm (–M) and 24.0 μm ( + M) around t = 350 nm in Fig. 9(b)). The large damping caused by the Fe governs the propagation loss more than the ridge dimension unless the mode field leaks out of the ridge. Although the propagation length is slightly improved compared with that for W = 1 μm, a reduction in the losses from the Fe entails a reduction in the NRPS (Re[Δneff]) and NRL.
Finally, it is worthwhile to investigate the performance of the LR-DLSPPW configurations for different refractive indices. We change the refractive index of the buffer layer from the default BCB (nb = 1.535) to higher (nb = 1.75) or lower (nb = 1.44) than that of the PMMA ridge (1.493), whose indices are respectively similar to Al2O3 and SiO2 [36]. The calculations are conducted, assuming the LR-DLSPPW configuration with the Fe stripe of WFe x TFe = 500 x 50 nm and the ridge dimension of W x T = 1 x 1 μm. The calculation results are shown in Fig. 11.
Fig. 11 The dependence of the nonreciprocal propagation characteristics on the refractive index of the buffer layer (nb) in the LR-DLSPPW configuration (WFe x TFe = 500 x 50 nm, W x T = 1 x 1 μm). (a) Re[neff] (red and blue solid curves, left axis) and Re[Δneff] (dashed black curves, right axis). (b) Lspp (red and blue solid curves, left axis) and NRL (dashed black curves, right axis). The upper insets show the field distributions of the norm of the electric field (TM-like) under the positive magnetization. The aspect ratio of the insets is different from reality for an illustration purpose. The white curve inside indicates the intensity along the vertical direction at the horizontal center.
With the higher refractive index of nb = 1.75, the balance of the mode field on either side of the Fe is achieved at even thinner buffer layer of t ≈100 nm than that with BCB as indicated by the peak in the propagation length (Fig. 11(b)). Since the refractive index of the buffer layer is much higher than that of the ridge, the mode field is no longer supported by the ridge but laterally radiative inside the buffer layer above t = 250 nm. Until this thickness, the buffer layer squeezes the mode field to the Fe surface because of its higher index, enhancing the NRPS (Re[Δneff]) and NRL while making the maximum propagation length shorter (Lspp = 20.5 μm (–M) and 19.0 μm ( + M) around t = 100 nm) than that for the lower index.
With the lower refractive index of nb = 1.44, it is observed that the buffer layer cannot sustain the balanced mode field on account of the too small refractive index and the propagation length increases monotonously without local maximum as the thickness of the buffer layer increases, where the light is distributed largely inside the buffer layer. That is why the NRPS (Re[Δneff]) and NRL lie lower than those of the higher index buffer layers.
The influence of the thickness and width of the ridge (W x T) and the refractive index of the buffer layer (nb) on the consequential figures of merit is summarized in Fig. 12, in which each trajectory follows the change in the thickness of the buffer layer from t = 40 nm to 500 nm.
Fig. 12 Parametric plots of (a) απ/2 NRPS versus Lπ/2 NRPS and (b) α1-dB NRL versus L1-dB NRL for the W x T = 1 x {0.7, 1.3} μm (red curves), W x T = {0.7, 1.3} x 1 μm (blue curves), and W x T = 1 x 1 μm with different refractive indices of the buffer layer (green curves), as compared with W x T = 1 x 1 μm with BCB buffer layer (black curve). The arrows indicate the direction of each trajectory as the thickness of the buffer layer changes from t = 0.04 μm to 0.5 μm.
Interestingly, it turned out that the minima of απ/2 NRPS and α1-dB NRL are almost limited to ~450 dB and ~10 dB, respectively, regardless of the ridge dimension or the refractive index of the buffer layer. The reduction in the ridge thickness (inversed open red triangles) and width (inversed open blue triangles) can shorten the required length, Lπ/2 NRPS and L1-dB NRL, owing to the enhanced TMOKE by the field confinement in the small ridge, while the incurred loss is increased. The increase in the refractive index of the buffer layer (nb = 1.75, filled green triangles) can also largely reduce Lπ/2 NRPS and L1-dB NRL, whereas again it is accompanied by much higher loss. As has been repeatedly discussed, these are attributed to the tradeoff of the MO effect and propagation loss related to the SPPs’ field confinement to the Fe layer. We put a priority on α1-dB NRL rather than απ/2 NRPS because of a realistic standpoint and conclude that the optimal dimension of the LR-DLSPPW configuration are WFe x TFe = 500 x 50 nm and W x T = 1 x 1 μm around t = 400 nm (indicated by A and B in Fig. 12).
7. Comparison and outlook
From all results presented above, the optimal dimension of the PMMA dielectric ridge of the nonreciprocal DLSPPW configuration is W x T = 1 x 0.47 μm for the minimum απ/2 NRPS = 357 dB and W x T = 0.3 x 0.17 μm for the minimum α1-dB NRL = 31.0 dB. The ridge dimension favorable for both απ/2 NRPS and α1-dB NRL is around W x T = 1 x 1 μm (απ/2 NRPS = 380 dB and α1-dB NRL = 31.4 dB).
For the nonreciprocal LR-DLSPPW configuration, the optimal dimension of the Fe stripe and the dielectric ridge are WFe x TFe = 500 x 50 nm and W x T = 1 x 1 μm, respectively, with the BCB buffer layer around t = 400 nm. Even though the required length depends on a combination of materials for the dielectric ridge and the buffer layer and the dimension of the ridge, the figures of merit are limited to απ/2 NRPS ≈450 dB and α1-dB NRL ≈10 dB at lowest.
The LR-DLSPPW configuration is slightly ahead of the DLSPPW configuration in terms of α1-dB NRL due to a reduction in the propagation loss. When it comes to απ/2 NRPS, however, the DLSPPW configuration is better than the LR-DLSPPW configuration because of the larger MO effect. The nonreciprocal performances are summarized in Table 1, compared with some previously reported magneto-plasmonic / semiconductor waveguide optical isolators.
Table 1. Comparison of nonreciprocal performances.
In comparison with other MO-SPP waveguides, it is possible that a 1D design is subject to deterioration of the nonreciprocal performances when it is adapted to actual 2D or 3D configurations. One has to take into account this fact as well as the material and structural compatibilities with the PICs process for practical designing. Nevertheless, the NRPS and NRL of the Fe DLSPPW/LR-DLSPPW are larger than those of photonic waveguides by virtue of the large MO constant of Fe and the field concentration on the Fe by SPPs. Easing the propagation loss of SPPs with keeping a sizable MO effect should be further pursued.
From the fabrication point of view, growing single-crystal Fe with smooth surface surely minimizes the propagation loss and maximizes the magnetic property. There have been studies on loss compensation by introducing optical gain into the dielectric ridge [43] or the buffer layer [44]. If those cases are applied, the propagation loss would be largely cured in view of the SOA isolator cases [31,32] in Table 1. Within passive modifications, combining noble metals with ferromagnetic metals has been eagerly studied [45], in which systems the MO effect is endorsed by the ferromagnetic metals whereas relatively lower losses of SPPs are warranted by the noble metals. In addition, using a trilayer structure consisting of double-dielectrics of high and low refractive indices and a ferromagnetic metal shows significant reduction of the propagation loss and the enhancement of the MO effects at the cutoff condition of SPP mode [10,11,46]. Studies on utilizing these structures must be of great interest for performance improvement.
The magneto-plasmonic waveguides can be applicable not only to the optical isolators but also to any conversion from magnetic signals to optical signals. Recently, an ultrafast spin-photon memory has been proposed, in which data is memorized in a magnetization state of a ferromagnetic metal by a circularly-polarized light pulse [47]. For the data reading, magnetization dependent loss (NRL) can be used. In that case, only 0.1-dB isolation seems enough to read off the data [48]. Our LR-DLSPPW configuration is capable of 0.1-dB NRL for the propagation loss of ~1 dB within a ~5.6 μm long waveguide.
Using ferromagnetic metals as MO materials, one should always take care of oxidation and corrosion. When covering with a passivation film, such as Al2O3, the optimal structure design might change. One can optimize the waveguide design to maximize the figure of merit by the procedure above. Enclosing the ferromagnetic metal with thin Au film such as the aforementioned noble and ferromagnetic metals hybrid structure [45] would also help this concern.
8. Conclusion
We have theoretically investigated magneto-plasmonic waveguides of the DLSPPW and the LR-DLSPPW configurations equipped with a ferromagnetic-metal Fe instead of noble metals. The nonreciprocal performances were evaluated by the FDFD method in terms of the nonreciprocal phase shift (NRPS) and nonreciprocal propagation loss (NRL) upon the magnetization reversal. The NRPS and NRL of the DLSPPW configuration are larger than those of semiconductor optical isolators by virtue of the large MO constant of Fe and the field confinement by SPPs although the accompanying loss is too large to work in practice. In order to reduce the loss incurred, the LR-DLSPPW configuration was introduced. We have optimized the device configuration aiming to minimize the propagation loss in return for the nonreciprocal effects. With the Fe layer of a thickness of 50 nm and a width of 500 nm between the 400-nm BCB buffer layer and the 1 x 1-μm PMMA dielectric ridge, the propagation loss for NRL of 1 dB is suppressed to ~10 dB within a ~56 μm long waveguide.
Even though the performances of the proposed magneto-plasmonic waveguides are still in early stage and not as good as immediately feasible for nonreciprocal devices such as optical isolators, they are still useful for the detection and conversion of magnetic signals into optical signals and envisage development by introducing optical gain media and/or by incorporating with thermo- and electro-optic effects and so on, thanks to the scalability that the DLSPPW and LR-DLSPPW configurations potentially have.
Our comprehensive investigation offers practical information on the magneto-plasmonic waveguides in PICs and how much nonreciprocal performance are expected and becomes the stepping stone to explore further functionalities that the magneto-plasmonics provides for on-chip photonics and electronics.
Funding
Grant-in-Aid for Scientific Research (Nos. 24656010 and 15K13942) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan
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