Study on the crucial conditions for efficient third harmonic generation using a metal-hybrid-metal plasmonic slot waveguide

Tingting Wu; Perry Ping Shum; Yunxu Sun; Xuguang Shao; Tianye Huang

doi:10.1364/OE.23.000253

1. Introduction

It is known that harmonics generation is a very practical way to produce new wavelengths of light. Particularly, third harmonic generation (THG) which was developed by Armstrong et al. [1] is attracting considerable attention due to its various applications, such as signal processing [2], material processing [3], scanning microscopy [4], and optical performance monitoring [5]. THG has been investigated both theoretically and experimentally in a number of different devices [6–11] since the pioneering observation in a silica fiber [12]. Nevertheless, rare works have been done addressing on the specific crucial conditions for efficient THG. On the other hand, plasmonic waveguides are ideal for nonlinear optical frequency conversion due to their extraordinary abilities of spatially confining electric field in subwavelength scale [13]. Another advantage of utilizing plasmonic waveguides is that the strength of different nonlinear terms can be engineered freely through employing the waveguide geometric parameters. A variety number of plasmonic waveguide structures have been proposed and investigated [14–18], but only waveguide of the metal-insulator-metal (MIM) type can provide true subdiffraction modal confinement despite the large linear propagation loss [18].

In this paper, we propose a nonlinear metal-hybrid-metal asymmetric plasmonic slot waveguide (MHM) structure to develop a method for high THG conversion efficiency with optical fields greatly confined in the subwavelength slot region. This waveguide can be considered analogous to the MIM. The ‘hybrid’ here is different from the traditional hybrid definition [19]. In particular, we define hybrid here because two different nonlinear materials are used to fill the metallic slot. It is found that, utilizing this structure, we are able to obtain efficient THG from mid-IR to near-IR regions within a short waveguide length, which would be of great importance in future nanophotonic modulations. To make an in-depth recognition of the THG process in this MHM waveguide, the influence of the geometric parameters to several key factors is analyzed in detail. Note that, we investigate efficient THG from mid-IR to near-IR regions in the MHM as an example because of the mature development of the compact, low-cost coherent mid-IR sources [20–29], the wide mid-IR spectral range applications in the telecom-band for optical communications [30, 31] and the quantum cascade lasers in the mid-IR waveband with superior performance [32, 33]. THG in the MHM can convert lasers at other wavelengths to shorter wavelengths, such as the infrared lasers to visible and near ultraviolet wavelengths.

2. Crucial conditions for efficient THG and waveguide structure

To model the THG process in lossy waveguides, the nonlinear coupled mode equations describing the interaction between the fundamental wave and the third-harmonic wave are used [34]. Light travelling in waveguides can be described with Maxwell’s curl equations,

\nabla \times \vec{E} (\vec{r}, t) = - μ_{0} \frac{\partial \vec{H} (\vec{r}, t)}{\partial t}

\nabla \times \vec{E} (\vec{r}, t) = ε \frac{\partial \vec{E} (\vec{r}, t)}{\partial t} + \frac{\partial {\vec{P}}_{N L}}{\partial t}

in which

ε = ε_{0} ε_{r}

, ε₀ and 𝜇₀ are the linear permittivity and permeability in vacuum, ε_r is the relative permittivity.

{\vec{P}}_{N L}

is the nonlinear polarization vector and

\partial {\vec{P}}_{N L} / \partial t

can be seen as a source term. We assume there are only two different propagating modes, i.e. the fundamental wave (FW) and the third harmonic (TH), and neglect the linear propagation losses at first, the total electric and magnetic fields at z waveguide location can be expressed as [35, 36]:

\vec{E} (\vec{r}, t)= \sum_{j} \frac{1}{2} {\tilde{A}}_{j} (z) Z_{0}^{1 / 2} {\vec{F}}_{j} ({\vec{r}}_{⊥}) \exp [i (β_{j} z - ω_{j} t)] + c . c .

\vec{H} (\vec{r}, t)= \sum_{j} \frac{1}{2} {\tilde{A}}_{j} (z) Z_{0}^{- 1 / 2} {\vec{G}}_{j} ({\vec{r}}_{⊥}) \exp [i (β_{j} z - ω_{j} t)] + c . c .

in which

j = 1

refers to FW,

j = 3

refers to TH,

{\tilde{A}}_{j} (z)

are the slowly-varying mode amplitudes,

{\vec{F}}_{j} (r_{⊥})

and

{\vec{G}}_{j} (r_{⊥})

are the mode profiles which are normalized by

\frac{1}{4} \iint_{A_{N L}} ({\vec{F}}_{j} \times {\vec{G}}_{j}^{*} + {\vec{F}}_{j}^{*} \times {\vec{G}}_{j}) \cdot \hat{z} d x d y = 1

, and

β_{j} = \frac{ω_{j}}{c} n_{e f f} (ω_{j})

are the propagation constants.

Z_{0} = \sqrt{μ_{0} / ε_{0}}

used here is to simplify the numerical calculation.

\vec{r} =(x, y, z)

and

{\vec{r}}_{⊥} =(x, y)

. With the normalization, the corresponding fields power can be described as

P_{j} (z) = {| A_{j} (z) |}^{2}

.

Applying the reciprocity theorem we can obtain:

\frac{d {\tilde{A}}_{j}}{d z} = - \frac{Z_{0}^{1 / 2}}{2} {\iint_{A_{N L}} 〈 \exp [- ​ i (β_{j} z - ω_{j} t)] {\vec{F}}_{j}^{*} \cdot \frac{\partial {\vec{P}}_{N L}}{\partial t} 〉}_{t} d x d y

where

{〈 \cdot \cdot \cdot 〉}_{t}

is time averaging,

A_{N L}

is the cross-section area of the waveguide, and the nonlinear polarization vector is described as [37]:

{\vec{P}}_{N L} = ε_{0} χ^{(3)} (\vec{r}) (\vec{E} (\vec{r}, t) \cdot \vec{E} (\vec{r}, t)) \vec{E} (\vec{r}, t)

For lossy waveguides, the complex propagation constant can be written into its real and imaginary parts as: $β = β_{j} + i α_{j} / 2$ . Therefore we define

A_{j} = {\tilde{A}}_{j} \exp (- \frac{α_{j} z}{2})

and substitute it into Eq. (6), the nonlinear polarization in lossy waveguide case can be written as:

{\vec{P}}_{N L}^{'} = ε_{0} χ^{(3)} (\vec{r}) ({\vec{E}}^{'} (\vec{r}, t) \cdot {\vec{E}}^{'} (\vec{r}, t)) {\vec{E}}^{'} (\vec{r}, t)

where

{\vec{E}}^{'} (\vec{r}, t)= \frac{1}{2} \sum_{j} A_{j} (z) Z_{0}^{1 / 2} {\vec{F}}_{j} ({\vec{r}}_{⊥}) \exp [i (β_{j} z - ω_{j} t)] + c . c .

. We then get:

\frac{d {\tilde{A}}_{j}}{d z} = - \frac{Z_{0}^{1 / 2}}{2} \exp (\frac{α_{j} z}{2}) {\iint_{A_{N L}} 〈 \exp [- ​ i (β_{j} z - ω_{j} t)] {\vec{F}}_{j}^{*} \cdot \frac{\partial {\vec{P}}_{N L}^{'}}{\partial t} 〉}_{t} d x d y

By substituting Eq. (7) into Eq. (9), we finally obtain the following set of nonlinear coupled-mode equations describing the THG process in lossy waveguides:

\frac{\partial A_{1}}{\partial z} = - \frac{α_{1}}{2} A_{1} + i [(I_{1} {| A_{1} |}^{2} + I_{2} {| A_{3} |}^{2}) A_{1} + I_{3} {(A_{1}^{*})}^{2} A_{3} e^{i δ β z}]

\frac{\partial A_{3}}{\partial z} = - \frac{α_{3}}{2} A_{3} + i [(I_{4} {| A_{1} |}^{2} + I_{5} {| A_{3} |}^{2}) A_{3} + I_{6} {(A_{1})}^{3} e^{- i δ β z}]

where

α_{j}

are the linear propagation loss coefficients, δβ is the phase-mismatch constant, and the nonlinear parameters

I_{1}

,

I_{2}

,

I_{3}

,

I_{4}

,

I_{5}

and

I_{6}

which are related to the modal overlap integrals between the fundamental and third harmonic waves are defined as:

I_{1} = \frac{1}{16} k_{1} Z_{0} χ^{(3)} (ω_{1}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} (2 {| {\vec{F}}_{1} |}^{4} + {| {\vec{F}}_{1}^{2} |}^{2}) d S

I_{2} = \frac{1}{8} k_{1} Z_{0} χ^{(3)} (ω_{1}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} ({| {\vec{F}}_{1} |}^{2} {| {\vec{F}}_{3} |}^{2} + {| {\vec{F}}_{1} \cdot {\vec{F}}_{3} |}^{2} + {| {\vec{F}}_{1} \cdot {\vec{F}}_{3}^{*} |}^{2}) d S

I_{3} = \frac{3}{16} k_{1} Z_{0} χ^{(3)} (ω_{1}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} ({\vec{F}}_{1}^{*} \cdot {\vec{F}}_{3}) ({\vec{F}}_{1}^{*} \cdot {\vec{F}}_{1}^{*}) d S

I_{4} = \frac{3}{8} k_{1} Z_{0} χ^{(3)} (ω_{3}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} ({| {\vec{F}}_{1} |}^{2} {| {\vec{F}}_{3} |}^{2} + {| {\vec{F}}_{1} \cdot {\vec{F}}_{3} |}^{2} + {| {\vec{F}}_{1} \cdot {\vec{F}}_{3}^{*} |}^{2}) d S

I_{5} = \frac{3}{16} k_{1} Z_{0} χ^{(3)} (ω_{3}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} (2 {| {\vec{F}}_{3} |}^{4} + {| {\vec{F}}_{3}^{2} |}^{2}) \cdot n_{0}^{2} (ω_{3}, {\vec{r}}_{⊥}) d S

I_{6} = \frac{3}{16} k_{1} Z_{0} χ^{(3)} (ω_{3}, {\vec{r}}_{⊥}) \underset{A_{N L}}{∯} ({\vec{F}}_{1} \cdot {\vec{F}}_{3}^{*}) ({\vec{F}}_{1} \cdot {\vec{F}}_{1}) d S

where

χ^{(3)} (ω_{j}, {\vec{r}}_{⊥})

are the third-order nonlinear susceptibilities at at FW and TH at any location of the waveguide and

k_{1} = 2 π / λ_{1}

is the wave number. Note that the THG nonlinear process is based on the third-order nonlinear susceptibility χ⁽³⁾ which also leads to TPA.

n_{2} (ω_{j}, {\vec{r}}_{⊥})

and

α_{2} (ω_{j}, {\vec{r}}_{⊥})

are the nonlinear refractive indices and two-photon absorption (TPA) coefficients which are related to the real and imaginary parts of χ⁽³⁾ by

\frac{ω}{c} n_{2} (ω_{j}, {\vec{r}}_{⊥}) + \frac{i}{2} α_{2} (ω_{j}, {\vec{r}}_{⊥}) = \frac{3 ω_{j}}{4 ε_{0} c^{2} n_{0}^{2} (ω_{j}, {\vec{r}}_{⊥})} χ^{(3)} (ω_{j}, {\vec{r}}_{⊥})

[38], where

n_{0} (ω_{j}, {\vec{r}}_{⊥})

are the linear refractive indices.

From Eqs. (10) and (11) we can observe that, efficient THG in lossy waveguides depends on three crucial conditions which can be provided separately as following:

1) The PMC must be satisfied.
2) The interactive material must be with high third-order nonlinear susceptibility χ⁽³⁾.
3) The pump-harmonic modal overlap must be sufficiently large in conjunction with reasonable linear propagation loss.

Many different waveguide structures have been explored to achieve high THG efficiency. For instance, in the highly Germania-doped fibers, the advantage of THG is the long interactive length, but the main limitation is the small pump-harmonic modal overlap between the phase-matched modes [10]. Silica microfiber has been considered to be a good candidate to achieve PMC between HE₁₁ mode at FW and HE₁₂ mode at TH with much higher pump-harmonic modal overlap compared with that in highly Germania-doped fibers. However the short interactive length and the small third-order nonlinear susceptibility limit the achievable conversion efficiency [6, 7]. In plasmonic slot waveguides, the three crucial conditions can be fulfilled or optimized simultaneously by adjusting the waveguide geometrical parameters and employing special nonlinear materials.

It is known that, efficient THG is possible via inter-modal phase-matching technique in plasmonic waveguides to phase-match the fundamental pump wave to the generated higher-order third harmonic signal which experiences the same effective mode index. Ideally, we would like to use two fundamental modes because of their same symmetrical field distributions. However, it is difficult to achieve PMC between these two fundamental modes because the corresponding fundamental wave always has a lower effective index than the third harmonic wave. Therefore, the fundamental wave must phase-match to higher-order modes of the generated third-harmonic wave. Note that, the inter-modal phase-matching technique can be utilized to achieve PMC by employing the waveguide geometries. The optimal THG conversion efficiency at certain waveguide structure may require a small initial detuning constant δβ negatively offset from 0 to compensate for the phase-mismatch during the THG process.

With respect to the nonlinear interactive material, silicon nanocrytal (Si-nc) which possesses high third-order nonlinear susceptibility has stimulated extensive research interest on all-optical high-speed signal processing [39, 40]. Recently, optical switching has been demonstrated by means of a ring resonator based on Si-nc slot waveguide structure [41]. Other materials with an order of magnitude higher third-order nonlinear susceptibility have been reported [42, 43], however, they have not been demonstrated in planar subwavelength photonic structures yet.

To achieve efficient THG, in addition to the PMC and the material with high third-order nonlinear susceptibility, a large enough pump-harmonic modal overlap which can be utilized to assess the relative contribution of the two interactive modes is more essential. According to Eq. (17), the pump-harmonic modal overlap integral is influenced by the field distributions of the FW and the TH, especially the TH since the FW is always assumed to propagate at fundamental mode. Therefore, it is difficult to fully take advantage from the inter-modal phase-matching technique due to the low corresponding pump-harmonic modal overlap in common planar waveguide structures [37]. To enlarge the pump-harmonic modal overlap integral, one efficient way is to reduce the negative contribution of the higher-order mode at TH. Asymmetric plasmonic slot waveguide structures have been proposed to achieve this requirement [34, 44]. Due to the asymmetric waveguide structure, the symmetric field distribution of the higher-order mode at TH is greatly broken and the counteraction effect of its negative part during the calculation of the modal overlap integral can be reduced significantly. We would emphasize the significance of the asymmetric structure here because the pump-harmonic modal overlap integral in common symmetric planar waveguide structures is very small under PMC condition. For example, the first mode exhibits an inverse symmetry in a common symmetric planar waveguide which results in a nearly zero modal overlap with the symmetric fundamental mode.

According to the three crucial conditions for efficient THG discussed above, the schematic diagram of our designed MHM structure is shown in Fig. 1 with Si-nc and silicon integrated into the metallic slot region. The whole waveguide is surrounded by air. The width and height of the Si-nc slot are 5r and h, respectively. The silicon slot width is r. Metal is defined to be silver (Ag) due to its relatively low induced linear propagation loss and with a Lorentz-Drude permittivity dispersion given by $ε_{A g} = ε_{\infty} - f_{p} / [f (f + i γ)]$ , with $ε_{\infty}$ = 5, $f_{p}$ = 2175 THz, and γ = 4.35 THz [45]. Material dispersions for accurately modeling the refractive indices of Si-nc and silicon are taken from [46, 47]. The PMC and the corresponding pump-harmonic modal overlap integral related I₆ will be evaluated to explore the THG conversion efficiency in part 3.

Fig. 1 Cross-section view of the proposed metal-hybrid-metal asymmetric plasmonic slot waveguide (MHM).

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To couple light between a silicon wire waveguide and this MHM, J. Tian et al. proposed experimentally a plasmonic coupler structure consisting of two tapered silicon strip waveguides and a plasmonic slot waveguide, in which a 35% coupling efficiency was demonstrated [48].

3. Simulation results and discussion

It should be stressed that, we have excluded the THG process both at the metal surfaces and in the metal layers during our calculation though metal benefits from strong third-order nonlinearity at optical frequencies. Since the electric field is tightly confined in the slot region, does not penetrate deeply inside the metal layers and decays exponentially with distance from the metal-dielectric interfaces, the generation of the nonlinear response is inhibited accordingly. Therefore, the THG process in the silver layers or at the silver surfaces can be rationally neglected. To find the required modal phase-matching waveguide geometry, the commercial COMSOL software for full-vector finite-element mode analysis is used to study the characteristics of the guided modes. In this part we choose the fundamental mode at FW to phase-match the first mode at TH since the pump-harmonic modal overlap integrals for the other modes are very small due to the symmetry considerations. We first set the slot height h to be 20 nm and adjust the silicon slot width r to fulfill the PMC. Figure 2 shows the dependence of the effective mode indices of the two guided modes on the silicon slot width r for a free space FW wavelength of 3600 nm. The black solid and the red dash lines represent the obtained effective mode indices for the FW and TH, respectively. As can be seen, the intersection point indicates that phase-matched THG is possible at the silicon slot width of 152 nm. The corresponding effective mode indices of FW (λ1 = 3600 nm) and TH (λ3 = 1200 nm) are 3.10513 + 0.08988i and 3.10511 + 0.01609i respectively. Figures 3(a) and 3(b) give the corresponding dominant electric field Ey distributions for the fundamental mode at FW and the first mode at TH. For the two guided modes at FW and TH, electric fields are tightly confined in the subwavelength metallic slot region as a result of discontinuity in the electric field normal to the metal-dielectric interfaces. It is worth to stress that, the first mode at TH results from the coupling between the fundamental mode in the Si-nc slot region and the opposite fundamental mode in the silicon slot region which forces the counteractive electric field distribution into the silicon part. This essential field-guiding mechanism differentiates the proposed MHM from previously reported MIM waveguides. Due to the special geometry of the silicon layer, the field confinement and enhancement of the opposite fundamental mode in silicon slot region are greatly reduced compared with the field in Si-nc slot. Therefore, the counteract contribution of the first mode at TH to the pump-harmonic modal overlap is small enough, which will surely results in high corresponding THG conversion efficiency. The principle of the sufficient large modal overlap in the proposed MHM can be seen more clearly in Fig. 3(c), where plots of the 1D normalized Ey distributions along the x cutline of y = 0 [as the horizontal dash lines plotted in Figs. 3(a) and 3(b)] are shown. For the considered MHM waveguide here, the negative part of the first mode at TH becomes almost negligible in the integration domain, thereby resulting in very small counteraction effect which in turn contributes to great enhanced pump-harmonic modal overlap as depicted in Eq. (17).

Fig. 2 Effective mode indices of the fundamental mode at FW and the first mode at TH as a function of the silicon slot width r in which the slot height is fixed to be h = 20 nm. ‘0’ stands for the fundamental mode, while ‘1’ stands for the first mode.

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Fig. 3 Electric field distributions of (a) fundamental mode at FW, (b) first mode at TH; (c) 1D normalized electric field distributions along the x cutline of y = 0 [as the horizontal dash lines shown in (a) and (b)].

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Then we carried out an optimization of the waveguide geometrical parameters in order to achieve the largest pump-harmonic modal overlap integral related nonlinear parameter I₆ in the proposed lossy MHM. For silicon, n₂(Si, 3600 nm) = 4.5 × 10⁻¹⁸ m²/W and n₂(Si, 1200 nm) = 7 × 10⁻¹⁸ m²/W [49], $α_{2} (S i) = 5.3 \times 10^{- 12}$ m/W is the TPA coefficient at TH [35]. While for Si-nc, $n_{2} (S i - n c)$ = 4.8 × 10⁻¹⁷ m²/W, $α_{2} (S i - n c) = 7 \times 10^{- 11}$ m/W at TH [50, 51]. The TPA coefficients at FW are set to be 0 for both Si-nc and silicon. The silicon is essentially linear loss-free during the calculation since the frequency of the incident radiation has insufficient energy to transfer electrons to the conduction band and cause linear absorption in the transmission window from 1.2 to 3.6 𝜇m [52]. Typically, it is valid to assume the nonlinear coefficient of Si-nc to be constant over the wavelength range from 1200 nm to 3600 nm [53]. We first analyze the influence of the slot height h on the pump-harmonic modal overlap performance at PMCs. In Fig. 4(a), the silicon slot width to satisfy the PMC and the corresponding $| I_{6} |$ are plotted as a function of the slot height. It is worth noting that, due to the complex mode fields and the existence of the TPA effects, the modal overlap integral related I₆ is complex as shown in Eq. (17). According to our calculation, only increasing the absolute value of the imaginary part of I₆ can enlarge the THG conversion efficiency. Therefore, we evaluate the absolute value of I₆, i.e. $| I_{6} |$ . As can be seen from Fig. 4(a), when the slot height increases, the silicon slot width required to ensure the PMC increases and $| I_{6} |$ decreases correspondingly due to the weaker field confinement. With an increase in the waveguide height, the electric field distribution at TH gradually distorts horizontally with more and more electromagnetic energy concentrates in the silicon slot region, which significantly reduces the pump-harmonic modal overlap. Then it can be summarized that the MHM has better THG property with narrower slot. However, too narrow slots cannot be considered because the fabrication feasibility should be taken into consideration. On the other hand, reducing the slot height can also increase the linear propagation loss induced by the metal layers. Since less power of FW can be transferred to TH with much higher absorption, we define the figure-of-merit (FOM) for the phase-matched THG process to evaluate the waveguide performance as FOM_FW, TH = $| I_{6} |$ /α_{FW, TH}, where α_FW and α_TH are the linear propagation loss coefficients at FW and TH, respectively. The FOMs as a function of the slot height are plotted in Fig. 4(b). Both FOMs at FW and TH decrease with increasing the slot height, which indicates that the most promising slot height for efficient THG is truly 20 nm, with the corresponding r = 152 nm, $| I_{6} |$ = 11450 m⁻¹W⁻¹, α_FW = 1.363 dB/𝜇m, α_TH = 0.73 dB/𝜇m, FOM_FW = 0.0084 W⁻¹dB⁻¹ and FOM_TH = 0.0157 W⁻¹dB⁻¹.

Fig. 4 (a) Silicon slot width r along with the modal overlap related $| I_{6} |$ , and (b) FOMs of FW and TH as a function of the slot height at different PMCs.

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Now we calculate the THG conversion efficiency by numerically solving Eqs. (10) and (11) in MATLAB for the case of Fig. 3 in which the fundamental mode at FW is converted into the first mode at TH. The THG conversion efficiency is defined as $η = P_{3} (L p) / P_{1} (0)$ for the lossy third-order interaction, where $P_{1} (0)$ is the fundamental pump power injected into the MHM, $P_{3} (L p)$ is the maximum output power of the generated third harmonic wave and $L p$ is the corresponding waveguide length. Fixed the incident power at FW to be 1 W. The inset of Fig. 5(a) reports the conversion efficiency contour map with different initial detuning constants. It is found that a maximum THG conversion efficiency is achieved at an optimized detuning of δβ = −9500 m⁻¹. This optimized initial negative detuning constant δβ is used to compensate for the nonlinear phase-shift during the THG process as mentioned in part 2. It has been demonstrated experimentally achievable by perturbing the waveguide geometry with pump-harmonic modal overlap almost unchanged in [54]. Specifically, a variation of 1 nm to the MHM slot width 6r (sum of the silicon slot width r and the Si-nc slot width 5r) corresponds to a detuning δβ variation of ± 2.68 × 10⁴ m⁻¹. The optical power evolutions of FW and TH waves along the propagation distance are graphed in Fig. 5(b) with an input pump power at FW fixed at 1 W and an optimized detuning of −2.68 × 10⁴ m⁻¹. It is seen that a peak power of the generated TH up to 2.79 × 10⁻⁴ W is revealed at a waveguide length of only 4.5 𝜇m, where the corresponding conversion efficiency is 2.79 × 10⁻⁴. We further calculate the conversion efficiency with different waveguide geometries to verify that the geometry optimized waveguide structure can truly be used to generate the most efficient THG. Figure 5(c) shows that the conversion efficiency with respect to the slot height at different PMCs with fixed pump power of 1 W and initial detuning of 0. It is obviously found that slot height of 20 nm under the PMC can lead to higher conversion efficiency. Fix the slot height to be 20 nm with initial detuning δβ = 0, the maximum output TH powers with different input pump powers are also investigated as illustrated in Fig. 5(d). The cubic function of the output TH power represents a significant advantage of THG in terms of nonlinear signal processing [2], material processing [3], scanning microscopy [4], and optical performance monitoring [5].

Fig. 5 Fixed the pump power to be 1 W, (a) contour map of the conversion efficiency with different initial detuning constants, (b) optical power evolutions of FW and TH along the propagation distance with the optimized detuning of – 9500m⁻¹, (c) conversion efficienncy and the corresponding detuning as a function of the slot height, (d) maximum output power P₃(Lp) versus the pump power of FW.

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Therefore, fusing the three crucial conditions of the PMC, high third-order nonlinear susceptibility of the interactive material and sufficient large pump-harmonic modal overlap, efficient THG in the lossy MHM is achievable with a small input pump power and an optimized initial detuning constant at short waveguide length.

4. Conclusion

We have reported three crucial conditions dependence of efficient third harmonic generation (THG) in lossy waveguides. In addition to the modal phase-matching condition (PMC) and high third-order nonlinear susceptibility of the silicon nanocrytal (Si-nc), large enough pump-harmonic modal overlap along with moderate linear propagation loss in the proposed metal-hybrid-metal asymmetric plasmonic slot waveguide (MHM) are achieved, which result in efficient THG. In the geometry optimized MHM, we get a THG conversion efficiency of η = 2.79 × 10⁻⁴ at the waveguide length of Lp = 4.5 𝜇m with input pump power of 1 W.

Acknowledgments:

This work was supported by the Basic Research Program of Shenzhen City (No. JCYJ 20140417172417146) and Key Laboratory of Network Oriented Intelligent Computation at Shenzhen graduate school, Harbin Institute of Technology. This work was also supported by the Singapore A*STAR SERC Grant: “Advanced Optics in Engineering” Program (Grant No. 1223600001).

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Study on the crucial conditions for efficient third harmonic generation using a metal-hybrid-metal plasmonic slot waveguide

Abstract

1. Introduction

2. Crucial conditions for efficient THG and waveguide structure

3. Simulation results and discussion

4. Conclusion

Acknowledgments:

References and links

Cited By

Figures (5)

Equations (17)

Optics Express