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We theoretically propose an internal asymmetric plasmonic slot waveguide (IAPSW), containing two different materials in the slot region. The IAPSW is used for second harmonic generation (SHG) at a wavelength of 1.55 μm. The required phase matching condition is satisfied between the 0th-order mode at the fundamental frequency and the 1st-order mode at the second harmonic frequency. By choosing appropriate slot geometry and materials, the mode field distribution is engineered to enhance the nonlinear coupling coefficient for SHG. With an 11 μm long IAPSW, a conversion efficiency of 24% (1.8 × 105 W−1cm−2 normalized conversion efficiency) is predicted. Furthermore, the SHG efficiency is more pronounced in IAPSW with thinner slot.
© 2016 Optical Society of America
1. Introduction
Second-order harmonic generation (SHG) known as a χ(2)-induced nonlinear process in optical material is attracting great attention because of its potential applications in the area of new wavelength generation [1], parametric amplification [2], and quantum photonics [3]. Benefiting from the highly compact size, tight mode confinement and tailored phase-matching conditions (PMCs), optical waveguide is a promising solution to realize efficient SHG. By using a silicon nitride (Si3N4) overlayer, an efficiency of 5 × 10−8 was achieved with 0.7 W peak pump power in a length of two millimeters waveguide [4]. The second harmonic nonlinearity can also be induced by static electric field. Based on this concept, combining the field enhancement in a ring resonator, an efficiency of 3.68 × 10−3 and 6.8 × 10−3 and was predicted with 75 mW and 60 mW pump power, respectively [5, 6].
More recently, plasmonic waveguides have attracted great attention because of their extraordinary abilities to allow strong confinement of light beyond the limits of diffraction in a dielectric medium [7, 8]. This unique property gives plasmonic waveguides an advantage for potential applications in the area of nonlinear photonics. Although the performances of plasmonic waveguides are currently limited by a trade-off between loss and mode confinement [9], they can still offer the potential to realize nonlinear photonics within very compact footprint [10–15]. Combining the second-order nonlinearity in lithium niobate (LiNbO3) and a plasmonic structure, some SHG waveguides have been proposed. However, the achieved conversion efficiency is in the order of 10−5 with 1 W pump [11–13]. Higher efficiency as large as 10−2 can be realized with a hybrid structure, but both mode confinement and propagation length are sacrificed [14]. Recently, dielectric-load plasmonic waveguide is proposed to realize broadband SHG, but the efficiency is still not high (~10−4) [15]. By employing a highly nonlinear polymer with an optimized waveguide structure, the SHG conversion efficiency can be significantly improved [16–18]. However, it should be noted that the concepts of hybrid plasmonic waveguide and hyperbolic waveguide proposed in [17, 18] pumped by Mid-IR source are not straightforward to implement near the C-band (1520 nm to 1570 nm). The materials silicon and germanium used in those devices show huge absorption at the harmonic wavelength (775 nm). However, all-optical signal processing is highly desired for optical communication system at C band. J. Zhang and et al. employed different materials to form the bottom and top of a vertical plasmonic slot waveguide [16]. This breaks the mode symmetry of the first-order mode at second harmonic wavelength. A conversion efficiency of 20% (normalized conversion efficiency of 1.3 × 105 W−1·cm−2) is predicted for the SHG conversion from 1550 nm to 775 nm. Because the symmetry is broken out of the slot region, here we name this type of waveguide as external asymmetric plasmonic slot waveguide (EAPSW). Although, in EAPSW, the symmetry of the first-order mode field distribution is broken to achieve nonzero mode overlap with the fundamental mode at wavelength of 1550 nm. Nevertheless, certain regions of the electric field distribution still contribute negatively (i.e. are counteractive) to the overall net nonlinear coupling coefficient (NCC) between the pump and harmonic modes.
In this paper, we propose an internal asymmetric plasmonic slot waveguide (IAPSW) for SHG. Like EAPSW, this waveguide is able to tightly confine the field in the sub-wavelength slot at both fundamental frequency (FF) and second harmonic frequency (SHF). Furthermore, two components including silicon nitride (Si3N4) and nonlinear polymer are chosen to be embedded in the slot region so that the NCC can be engineered to increase the conversion efficiency. The SHG performance in the IAPSW is then analyzed by numerical simulations.
2. Theory and design of the proposed IAPSW
The 3D and cross-section geometry of the proposed IAPSW is demonstrated in Fig. 1. The metallic slot with thickness h and width w formed by silver is filled with two different materials namely, Si3N4 and polymer with width w1 and w2, respectively. The considered polymer is the same as the one used in [16] with a second order nonlinear susceptibility of 619.4 pm/V. For silver, Drude permittivity dispersion of silver is given by εAg = ε∞ - fp2 / [(f (f + iγ)], with ε∞ = 5, fp = 2175 THz, and γ = 4.35 THz [19]. For Si3N4, the refractive indices of 2.1 and 1.9 are considered at 775 nm and 1550 nm, respectively. The substrate is formed by silica, and the whole device is surrounded by air.
First, the total width w and slot thickness h are fixed at 450 nm and 50 nm, respectively and the thicknesses of the bottom and up silver layers are set as 300 nm. By using COMSOL software for numerical calculation, the effective indices change of zeroth-order mode (labeled 0th-mode) at FF and first-order mode (labeled 1st-mode) at SHF with different Si3N4 width is demonstrated in Fig. 2(a). For PMC, the effective indices of the two interactive modes are required to be equal. According to the result, it is found that with w1 = 64 nm, the effective indices for FF and SHF are 2.2478 + 0.0071i and 2.2477 + 0.0060i, respectively. The normalized mode size and propagation length [20] are 0.017 and 17 μm for FF and 0.061 and 10 μm for SHF, respectively. Please note that the losses of non-epitaxial thin silver films can be much higher than the model we used. According to our calculation, when the thickness of silver layer is more than 300 nm, both the PMC geometry and imaginary parts of effective indices of two modes keep almost constant with respect to silver thickness variation. Therefore, in practice, thick silver film can be fabricated so as to avoid additional loss caused by too thin silver layers.
Fig. 2 (a) Effective indices 0th-mode at FF and 1st-order mode at SHF with respected to the width of Si3N4, (b) Ey profile of 0th-mode at FF, and (c) 1st-mode at SHF.
It should be noted that merely satisfying PMC is not sufficient for efficient SHG. High NCCs which are determined by the nonlinear susceptibility and mode overlap is another issue which must be paid attention to. The NCCs can be defined as follows [21]
where E1 and E2 are the normalized electric field of modes at FF and SHF, respectively.In this work, the polymer ensures a high second-order nonlinear susceptibility. On the other hand, since the slot region is composed by two different materials, the mode symmetry is broken leading to large mode overlap, as shown in Figs. 2(b) and 2(c). It is known that the electric field of 1st-mode at SHF possesses both positive part and negative part which have counteractive effect when overlapping with 0th-mode at FF as a result reducing the NCCs. However, according to Eq. (1), the NCCs are also weighted by χ(2). In our design, as shown in Fig. 2(c), a large portion of the negative electric field of 1st-mode at SHF is distributed in the Si3N4 region with negligible second-order nonlinearity and therefore the counteractive effect can be reduced. This is the main difference comparing with the EAPSW in which the mode symmetry is break by employing different materials to form the bottom and top of the slot. In an EAPSW, since the slot is only filled with nonlinear polymer, the main negative part of electric field of the 1st-mode is still in the highly nonlinear region and canceling a portion of mode overlap. For easy comparison, Figs. 3(a) and 3(b) demonstrate the main electric fields distributions of the EAPSW (Ex) and the IAPSW (Ey). It should be mentioned that, in a metal-insulator-metal (MIM) structure, only the field with polarization being perpendicular to the metal-insulator interface can be effectively confined within the slot. The slot is vertical in the EAPSW while horizontal in the IAPSW. Therefore Ex in EAPSW and Ey in IAPSW are presented in Fig. 3. It is found that, in an IAPSW, a large portion of the negative electric field is distributed in lower nonlinearity region namely Si3N4, while the negative peak is within polymer in an EAPSW. According to our calculation, the NCC in the proposed IAPSW is as high as 344 ps∙m−1∙W-1/2 comparing with the value of 294 ps∙m−1∙W-1/2 in the EAPSW with the same slot thickness of 50 nm.
Next, we change the total slot width w and adjust the Si3N4 width w1 to find different structures for PMC in an IAPSW with slot thickness fixed to be 50 nm. The results are demonstrated in Fig. 4. It can be found that with larger total width, the required Si3N4 width is increased. When the total width is reduced to 400 nm, no PMC geometry can be found. When the total width is in the range from 420 nm to 525 nm, the NCC values obtained in the IAPSW are larger than the value obtained in the EAPSW at PMC geometries.
3. SHG in the proposed IAPSW
In this section, we investigate the SHG performance in the proposed IAPSW by solving the couple-mode equation [16]
where AFF and ASHF are the mode amplitudes at FF and SHF, respectively. αFF and αSHF are the loss coefficients at FF and SHF, respectively. In this paper, the conversion efficiency is defined aswhere P2(L) is the power of second harmonic at waveguide length L, and P1(0) is the input pump power. Additionally, for the ease of comparison with EAPSW, we also use the definition of normalized conversion efficiency to quantify the SHG performance as [16, 22]where Lp is the length where the power of second harmonic reaches its maximum (defined as peak position), and P2(Lp) is the corresponding maximum output power of second harmonic. Figure 5(a) is a plot of FF and SHF waves as a function of propagation length with an input pump power of 1 W in phase-matched IAPSW (slot thickness of 50 nm, slot total width of 450 nm, and Si3N4 width of 64 nm). Due to the trade-off between loss and mode confinement in plasmonic waveguide, during the propagation, the SHF signal experiences both linear Ohmic loss caused by metal and nonlinear parametric gain induced by the pump. Initially, because of the strong pump, the power transfers from the FF to the SHF is sufficient to overcome Ohmic loss, and the power of SHF can be accumulated. Meanwhile, the pump wave at FF experiences not only linear Ohmic loss but also nonlinear loss, which accelerates its power decay. It can be found that the actual propagation length of FF reduces to be ~8 μm due to the SHG process. When the pump power becomes weaker, the nonlinear parametric gain obtained by SHF wave is not able to compensate the linear Ohmic loss, leading to power decay with longer waveguide. Therefore, the SHG power reaches its maximum of 0.24 W after propagating in an 11.4 μm short waveguide. The conversion efficiency is much higher than that of the previous works based on other types of plasmonic waveguides [11–15]. The normalized conversion efficiency 1.8 × 105 W−1∙cm−2 also demonstrates 38% increment comparing to the EAPSW at the same pumping level. On the other hand, because the optimized IAPSW possesses large NCC, efficient process can still happen at lower pump level. Figure 5(b) demonstrates the achievable peak conversion efficiencies and the required waveguide length at different pump level. Both the efficiency and length are strongly dependent on the input pump power. Considering a low power level of ~10 mW, the SHG efficiency is still as high as ~0.52% which is comparable with the resonance-enhanced silicon nitride ring resonator scheme with a 60 mW pump level [6].Fig. 5 Optical power evolution of FF and SHF along propagation distance with 1 W pump power, and (b) peak efficiencies and their corresponding positions at different pump levels.
In Fig. 6(a), the achievable peak efficiencies and their corresponding efficiency peaks at different PMC geometries under 1 W pump are demonstrated. Consisting with the trend of NCC, as shown in Fig. 3, the PMC geometry with largest NCC illustrates the highest conversion efficiency and the shortest required waveguide length. Generally, the achievable efficiencies are as high as ~20%. Comparing with EAPSW, both the achievable conversion efficiency and normalized conversion efficiency are even higher under PMC geometries with total slot width ranging from 420 nm to 500 nm, as shown in Figs. 6(a) and 6(b).
Fig. 6 (a) Peak efficiencies and their corresponding peak positions, and (b) Normalized conversion efficiency at different PMC point with 1 W pump power.
When considering fabrication process, the fabrication-error can induce unwanted phase mismatch into the device leading to efficiency reduction. To investigate the impact of phase mismatch, the conversion performance at different Δβ is plot as a contour map as shown in Fig. 7. The maximum conversion efficiency appears at Δβ = 0, and reduces with the absolute Δβ increment. To investigate the practical fabrication tolerance, we numerically calculate the waveguide with geometry deviations of ± 5 nm of slot thickness and ± 20 nm of total and Si3N4 widths. Within such tolerance range, we find the maximum phase mismatch is ~3.04 × 105 m−1, which may cause nearly 3 dB reduction of the achievable efficiency. Therefore, the proposed waveguide possesses a manageable fabrication-error tolerance.
In the above simulation, only the performance with fixed slot thickness of 50 nm is analyzed. In order to comprehensively compare EAPSW with IAPSW, we further simulate these waveguides to find PMC geometries under various slot thicknesses. For vertical slot in the EAPSW, each slot thickness corresponds to a specific slot height in order to satisfy the PMC, as shown in Fig. 8(a). However, the case is different for the IAPSW where the slot is filled with two materials. Even with fixed slot thickness, different combinations of these two materials will lead to multiple PMC geometries, as shown in Fig. 4. According to our calculation, only the PMC geometries corresponding to the best SHG performance at each thickness are shown in Fig. 8(b). It can be found that both the NCCs and losses for those two structures increase with the reduction of slot thickness, as shown in Figs. 8(c) and 8(d). However, the NCCs of IAPSW are always higher than that of EAPSW at same slot thickness. Under condition of 10 nm slot thickness, the NCC of IAPSW reaches 3820 ps∙m−1∙W-1/2, which is nearly 1.4 times that of EAPSW. Under 1 W pump power, the peak conversion efficiencies, efficiency peak positions, and normalized conversion efficiencies for those two structures are presented in Figs. 8(e)-8(h). Obviously, with thinner slot thickness, the superiority of IAPSW is more pronounced. For example, for the case of 10 nm slot thickness, conversion efficiency of 50% (normalized conversion efficiency of 2.55 × 107 W−1·cm−2) can be achieved within 1.4 μm long IAPSW, while the EAPSW can only realize conversion efficiency of 38% (normalized conversion efficiency of 1.3 × 107 W−1·cm−2) and the required waveguide length is 1.7 μm. Therefore, the proposed IAPSW is verified to be an effective solution to further increase SHG conversion efficiency, especially in the scenario of thinner slot.
Fig. 8 PMC geometries, NCCs and losses versus slot thickness in (a, c) EAPSW and (b, d) IAPSW. Under 1 W pump, the achievable peak SHG efficiencies, required waveguide lengths, and normalized conversion efficiencies versus slot thickness in (e, g) EAPSW, and (f, h) IAPSW.
In order to satisfy PMC, EAPSW only has two geometry parameters for optimization, including slot thickness (short edge length) and slot height (long edge length). According to our simulation, with a fixed slot thickness, only a specific slot height can be chosen for the purpose of PMC, as shown in Fig. 8(a). However, in IAPSW, even with fixed slot thickness and total slot width, the variation of Si3N4 width is helpful to achieve two PMC geometries. For example, as shown in Fig. 9(a), when the fixed slot thickness of 30 nm and slot width of 325 nm are chosen, the variation of Si3N4 width brings two PMC geometries. This feature is also possible with slot thickness of 40 nm and slot width of 375 nm, as shown in Fig. 9(b). Therefore, IAPSW provides more geometry choices for waveguide design and fabrication, which may be useful especially when other structural parameters are constrained.
Fig. 9 Effective indices of 0th-mode at FF and 1st-order mode at SHF versus Si3N4 width with (a) slot thickness fixed to be 30 nm and total slot width fixed to be 325 nm, and (b) slot thickness fixed to be 40 nm and total slot width fixed to be 375 nm.
Overall, although fabricating Si3N4/polymer layer sandwiching between two silver films requires additional fabrication steps comparing with EAPSW, the internal asymmetric structure provides a route for nonlinear mode overlap engineering, resulting in better SHG performance especially on the scenario of thin slot, and flexible choices for waveguide geometry design.
4. Discussion and conclusion
In order to excite the gap plasmonic mode in the IAPSW, a silicon strip waveguide can be placed beside the IAPSW for evanescent field coupling [23]. However, this method has not been experimentally realized. A practical and efficient method to couple light into the horizontal MIM waveguide is still a hot research topic. Additionally, when the evanescent mode coupling method is chosen, both the linear and nonlinear effects occurred at the coupling region should be considered [24]. Besides SHG devices, the concept of IAPSW can also be implemented in devices for other nonlinear processes such as intermodal third harmonic generation when the PMC is designed to satisfy between a symmetric mode and an asymmetric mode at pump and harmonic frequencies, respectively.
In summary, we have designed a silicon compatible nonlinear plasmonic slot waveguide with two materials (Si3N4 and polymer) infiltrated in the metallic slot region for SHG. The PMC for SHG in the proposed waveguide is achieved between the 0th-order mode at FF and 1st-order mode at SHF by properly optimizing waveguide geometry. In addition, by using two materials in the slot region, a large portion of counteractive part of the 1st-order mode electrical field can be forced into the low nonlinearity silicon nitride area resulting in high NCC for SHG. According to our numerical calculation, the SHG efficiency of up to 24% (normalized conversion efficiency up to 1.8 × 105 W−1cm−2) is predicted for the proposed waveguide with 1 W pump power. The conversion efficiency is still considerable under low pumping level (~10 dBm). Additionally, the conversion can be further enhanced by shrinking the thickness of the slot. These features make the proposed waveguide a promising candidate for integrated and power-efficient all-optical signal processing applications.
Acknowledgments
This work was partially supported by the Start-up Grant of China University of Geoscience (Wuhan) (Funding No. G1323511665) and 863 High Technology Plan (Funding No. 2015AA015502).
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