Optical Isolator Utilizing Surface Plasmons

Feasibility of usage of surface plasmons in a new design of an integrated optical isolator has been studied. In the case of surface plasmons propagating at a boundary between a transition metal and a double-layer dielectric, there is a significant difference of optical loss for surface plasmons propagating in opposite directions. Utilizing this structure, it is feasible to fabricate a competitive plasmonic isolator, which benefits from a broad wavelength operational bandwidth and a good technological compatibility for integration into the Photonic Integrated Circuits (PIC). The linear dispersion relation was derived for plasmons propagating in a multilayer magneto-optical slab.


Introduction
The magneto-optical (MO) effect is important for a variety of applications. A unique feature of the MO effect is non-reciprocity. The optical properties of non-reciprocal devices are different for two opposite directions of light propagation. The non-reciprocal effect can occur only in a MO material and the important optical non-reciprocal devices, such as an optical isolator and an optical circulator, can only be fabricated by utilizing MO materials. An optical isolator is an important component of optical communications systems. It protects optical elements from back-reflected light. The protection by the optical isolator is essential for the normal operation of optical components. For example, the OPEN ACCESS operation of high-speed optical amplifiers and high-speed laser diodes is impossible without protection by an optical isolator, because of the gain modulation by back-traveling light.
The performance of an optical isolator is characterized by 3 main parameters: the isolation ratio, the insertion loss and the operational wavelength bandwidth. At present, the bulk-type optical isolator is commercially available. The structure of the bulk-type optical isolator is rather simple. It consists of a magnetic garnet inserted between two polarizers. Since both magnetic garnets with low absorption and the polarizers with low insertion loss and high extinction ratio are available, the performance of the present commercial optical isolator is close to perfect. The optical bulk isolators with isolation ratio of 40 dB, insertion loss of 0.7 dB and bandwidth of 20 nm are commercially available.
The integration of optical elements into Photonic Integrated Circuits (PIC) is an important task, because it may reduce cost and improve performance of high-speed optical data processing circuits for the high-speed optical networks. The integration of an optical isolator is important for PIC, because the problem of back-reflected light is more severe in the case of integrated optical elements. However, the optical isolator is one of the few optical components, which has not yet been integrated into commercial PIC. At present, there is a strong industrial demand for the integrated optical isolator.
It is difficult to integrate the optical isolator into PIC for the following reason. The indispensable component of any optical isolator is a magneto-optical (MO) material. Traditionally, the magnetic garnets are used as a transparent MO material. However, it is difficult to grow the high-crystal-quality magnetic garnets on semiconductors substrates such as Si, GaAs, InP, which are used as a substrate for PIC, because the conventional growth temperature of the garnets is above the melting temperature of these substrates.
To overcome this problem several solutions have been proposed. A wafer bonding between garnets and semiconductors [1] and sputtering of garnet on Si [2] were successfully used to fabricate the isolator on a Si substrate. Also, diluted magnetic semiconductors CdMnTe and CdMnHgTe [3] have been used for the fabrication of optical isolator on a GaAs substrate. Even though a good performance of the above-mentioned standalone isolators fabricated on the semiconductor substrates were achieved, the integration of the isolators with other optical elements on the same substrate has not been demonstrated yet. The integration is difficult, because MO and optical properties of the garnets and diluted magnetic semiconductors (DMS) significantly depend on their crystal quality. As the crystal quality of garnets or DMS degrades, the MO constants in these materials are reduced and the optical loss is increased. Even if it is possible to keep a high crystal quality of the MO material during the fabrication of a standalone isolator, the crystal quality usually degrades during the fabrication of other elements of PIC. That is the reason why there are still difficulties to integrate a garnet-based isolator or a DMS-based isolator into a commercial PIC.
Ferromagnetic metals are attractive as an MO material for the optical isolator. They have very large MO constants and importantly they are technologically compatible with the fabrication technology of PIC. For the ferromagnetic metals the conventional sputtering and lift-off techniques may be used. They do not degrade during the fabrication of other optical components of PIC. There is one significant demerit of the ferromagnetic metals. They significantly absorb light. To overcome this problem, the design of an isolator, in which the optical absorption by a ferromagnetic metal is compensated by an optical gain of a semiconductor optical amplifier, has been proposed [4] and realized [5][6][7]. This type of isolator provides high isolation and has low optical insertion loss.
However, the required operational pump current for this isolator is very high. At present, it is about 100 mA. The required current is unacceptably high for PIC. That is a reason why this type of isolator has not been integrated into PIC as well.
Here we propose to utilize the surface plasmons propagating at the surface of the ferromagnetic metals to fabricate an efficient optical isolator.
It should be noticed that since the performance of the bulk isolator is nearly excellent, at present it is unfeasible that an isolator integrated into PIC would reach a similar performance. For the merit of the integration, some penalties should be paid. Therefore, some parameters of the integrated isolator should be compromised. For example, in the case when the operational wavelength bandwidth may be compromised, a ring-type isolator may be a good option [1,2]. The ring-type isolator has very narrow bandwidth of about 0.01 nm.
The surface plasmons intrinsically have a substantial optical loss. It is unlikely that the insertion loss and the isolation ratio of the plasmonic isolator would be the same as those of the bulk isolator. However, the ferromagnetic metals have very low wavelength dispersion of the optical and MO constants and the plasmonic isolator might be a good option for the integrated isolator when a wide operational wavelength bandwidth is required. As was mentioned above, the technological compatibility with PIC fabrication technology is another important merit of the plasmonic isolator.
MO effect for the surface plasmons was studied both theoretically and experimentally [8][9][10][11]. Here we study how to optimize the MO effect for the surface plasmons in order to fabricate an efficient optical isolator. For the optical isolator we are proposing to use single-surface plasmons propagating along a double-dielectric/ferromagnetic-metal interface. We have already shown [12] that it is possible to achieve a significant reduction of the optical loss and the enhancement of MO effect for surface plasmons propagating in this structure. We discuss MO properties of surface plasmons in the case when a magnetic field is applied perpendicularly to the plasmon propagation direction and in the film plane. In this case the unique properties of the transverse non-reciprocal MO (nMO) effect are utilized for MO enhancement [12]. The transverse nMO effect occurs in the case when light propagates perpendicularly to the magnetic field. In reference [12] we have shown that light can experience the transverse nMO effect only when it propagates in the vicinity of a boundary between two materials and the optical field at least in one material is evanescent. The transverse nMO effect occurs because the polarization of light, which has an evanescent component, rotates around an axis, which is parallel to the magnetic field and perpendicular to the light propagation direction. The magnitude of the transverse nMO effect is comparable to or often greater than the magnitude of the conventional longitudinal nMO effect. In contrast to the longitudinal nMO effect, the transverse nMO effect can be significantly magnified by optimizing the device structure.
A linear dispersion relation, which describes optical and MO properties of the surface plasmons propagating in single-dielectric, double-dielectric and multilayer structures, will be derived in Section 4.

Surface Plasmons
Surface plasmons have proved to be very efficient in localizing and guiding photonic signal on a scale which is hardly reachable with conventional dielectric waveguides. This makes them a prospective candidate for downsizing integrated optical circuits. A surface plasmon is an electromagnetic wave coherently coupled to electron oscillations and propagating in a wave-like fashion along a metal-dielectric interface. Currently, two types of the surface plasmons are utilized in optical applications [13]. A long-range surface plasmon propagates in a thin metallic film. A single-surface plasmon propagates along a boundary between a relatively-thick metallic film and a dielectric. Only the single-surface plasmons will be discussed in this paper. Figure 1a shows the optical intensity distribution for a plasmon propagating along metal-dielectric interface. Since light is tightly confined near the metal-dielectric interface, the plasmons are often used for the dense integration of optical elements or when an ultra dense focusing of light is required (for example, MO recording or medical applications). For the surface plasmons, optical field is distributed partially inside the metal and partially inside the dielectric. Since the metal absorbs light, the plasmons always experience optical loss. However, the plasmon's optical loss is significantly smaller than the optical loss for light, which is propagating through a bulk metal. This is due to the fact that in the case of plasmons a substantial amount of light energy propagates inside the dielectric and only a tiny amount propagates inside the skin depth in the metal. Particularly, the optical loss is small for plasmons in structures, which contain a low-resistance metal like Au, Cu, Al, Ag. Figure 1b shows 1/e propagation distance of plasmons in low-resistance metals Au and Cu. For these metals, the plasmon's 1/e propagation distance is long, about 200-500 μm. Because of low optical loss, mainly these metals are used for the variety of plasmonic applications. However, Au and Cu are not ferromagnetic metals and a ferromagnetic metal is required for the fabrication of an optical isolator. Figure 2a shows 1/e plasmon's propagation distance in the ferromagnetic metals Fe, Ni, Co. Since the resistance of these metals is higher, the plasmon's optical loss is also higher and the 1/e propagation distance is significantly shorter than in the case of the surface plasmons in Au and Cu. However, the problem of the high optical loss could be resolved by shortening device length. Since MO constants of the ferromagnetic metals are very large, even for a short propagation distance the MO FoM is a value, which numerically estimates the ability of the plasmons to provide the high isolation at required low insertion loss.  Figure 2b shows the MO FoM for plasmons propagating along Fe/air, Co/air, Ni/air, Co/AlGaAs interfaces. In all cases the FoM is about 2-8%. The MO FoM for the plasmons is comparable to MO FoM for a waveguide mode in optical waveguide covered by a ferromagnetic metal. As was discussed above this value might not be sufficient to fabricate an efficient plasmonic isolator. The data were calculated by solving Maxwell's equations (Section 4) and utilizing known optical and magneto-optical constants of metals [15][16][17] and semiconductors [18,19].
In the next section we will show that it is possible to reduce the optical loss and to increase the MO FoM in plasmonic waveguides using a dielectric, which consists of two materials with substantially different refractive indexes.
It is worth to mention two important facts about the surface plasmons. As can be noticed from Figure 2a, the optical loss of surface plasmons depends significantly on the refractive index of the dielectric. The higher the refractive index of the dielectric, the larger loss the plasmons experience. The second fact is that the effective refractive index [20] of surface plasmons is always very close to the refractive index of the dielectric. Figure 3 shows the effective index of surface plasmons propagating along Co/AlGaAs interface. The dashed lines show the refractive index of the corresponding dielectric. In all cases the plasmon's effective refractive index is only slightly larger than the refractive index of the dielectric.   The reason for the reduction of plasmon's optical loss near the cutoff is the following: When Al 2 O 3 thickness approaches a "cutoff' thickness, the penetration depth of the plasmon's optical field into silicon significantly increases, but the penetration depth into the iron remains unchanged. That means that the amount of the optical field decreases inside the metal and increases inside the silicon. Therefore, the optical field is "pushed out" from the iron. This causes a significant decrease of optical loss and increase of the plasmon's propagation distance. The following explains the increase of MO FoM near the cutoff thickness. A change of the refractive index of the metal changes the value of the "cutoff' thickness. The plasmon's optical loss sharply decreases when the Al 2 O 3 thickness approaches the "cutoff' thickness. Therefore, in the case of Al 2 O 3 thickness close to the "cutoff' thickness, a small MO change of the refractivity of the metal causes a significant MO change of the plasmon's optical loss.
As seen from Figure 4b, the MO FoM may reach 100%. This is a significant enlargement of MO FoM compared to 2-8% of the MO FoM, which is the most that can be achieved for the structure in which only one material of the dielectric is used (Figure 2b). It should be noted that the optical isolation and loss is significantly larger near the cutoff thickness of 16 nm than near the cutoff thickness of 400 nm, even though MO FoM near both cutoff thicknesses is comparable (see Figure 4b). Therefore, depending on the application, either a longer plasmonic isolator or a shorter plasmonic isolator can be made. For the longer plasmonic isolator, an Al 2 O 3 thickness of a little more than 400 nm should be used and for the shorter plasmonic isolator an Al 2 O 3 thickness a little less than 16 nm should be used.
As was shown above, there is a significant difference of optical loss for the surface plasmons propagating in a double-layer-dielectric plasmonic waveguide. It should be noticed that in a MO plasmonic waveguide it is not only the plasmon's absorption coefficient, which is dependent on a plasmon's propagation direction, but the effective refractive index depends on propagation direction as well. Therefore, a phase shift, which the plasmons experience after propagation through the MO plasmonic waveguide, is different for two opposite propagation directions. The non-reciprocal phase shift for waveguide modes in dielectric MO waveguide was used to fabricate an efficient MO isolator [1,2,14]. It is interesting to examine how large the non-reciprocal phase shift is for the plasmon and whether there is any enhancement of the non-reciprocal phase shift in the case of a double-dielectric plasmonic waveguide. Since the optical loss of the plasmons is not negligible, the non-reciprocal phase shift per given optical loss is important value for practical application. Therefore, another Figure- There is no enhancement of the non-reciprocal phase shift, but there is an enhancement of FoM for the non-reciprocal phase shift near the cutoff thicknesses, because of the loss reduction. Even though the enhancement of FoM for the non-reciprocal phase shift is smaller than the enhancement of FoM for the non-reciprocal loss, it is still prominent.

Dispersion Relation for Plasmons Propagating in Multi-Layer MO Slab
In the following we will derive the dispersion relation, which describes the MO effect for the surface plasmons propagating in a multilayer MO slab.
The plasmon's propagation direction is along the z-axis, the surface normal is along the x-direction and the magnetic field is applied along the y-direction ( Figure 6). For this direction of the magnetic field, the permittivity tensor for the ferromagnetic metal will be where both d ε and γ have non-zero real and imaginary parts.
will be 2 The optical field in j-layer can be described as where j b A and j f A are unknowns, x j is the coordinate of boundary between j and (j+1) layers and c.c. is complex conjugative. Introducing new unknowns the Equation (7) will be ( ) Simplifying Equation (9) Knowing the value of Z 1 the value of Z 2 can be calculated from Equation (13). Knowing the value of Z 2 the value of Z 3 can be calculated from Equation (13) Therefore, the values of Z for all layers can be calculated from Equation (12), if Z of one layer is known. T' j describes the variation of Z across j-layer, F j+1,j describes of change Z across the boundary between j and j+1 layers.
The Equation (12) can be solved for Z j as To derive plasmon's dispersion relation, we use the fact that the value of Z is known for the top and the bottom layers. In the case of a surface plasmon, light is confined near the metal/dielectric interface, therefore 0 E →  when x → ±∞ . For the bottom layer number 1, which is infinite when x → −∞ , Equation (6) will be Since Z 1 also can be calculated from known value of Z m using Equation (17)