Direct generation of graphene plasmonic polaritons at THz frequencies via four wave mixing in the hybrid graphene sheets waveguides

Yu Sun; Guofu Qiao; Guodong Sun

doi:10.1364/OE.22.027880

1. Introduction

Ultrahigh carrier mobility, excellent thermal stability, and unusual carrier-density-dependent surface conductivity of graphene make it a promising platform to build highly integrated plasmonic devices in THz and far-infrared (IR) frequencies [1, 2]. Graphene surface plasmons (GSPs) predicted by Jablan et al [3] have received experimental confirmation recently [4]. Compared to surface plasmon polaritons (SPPs) in noble metals, the GSPs manifest unique tunability using external gates, presumably long plasmon lifetime and deep sub-wavelength confinement [5]. The appealing properties of GSPs opens new ways for realizing plasmonic nano-sensors. transformation optics, metamaterials, and perfect absorbers [6].

However, a key challenge is the excitation of GSPs by external radiation due to the strong mismatch between wave numbers in graphene and free space [3]. The coupling of GSPs requires either near-field techniques using complex nano-antennas [7, 8] or far-field coupling employing periodical patterns on graphene sheets [4, 9], graphene strip [10, 11] and the substrates [12]. While the edge scattering could reduce the plasmon lifetime. Very recently, the sinusoidal diffraction gratings on graphene sheets generated by the surface acoustic waves [13] and the flexural waves [14] are presented. While such corrugations are excited by mechanical vibrators, which is not CMOS compatible.

The THz generation via nonlinear process in 2D waveguides are previously studied in the LiNbO₃ embedded waveguide [15], the LiNbO₃ ribbon waveguide [16] and the metallic slot waveguide [17] via difference frequency generation (DFG) or in the silica single mode fiber [18] and the silicon membrane waveguide [19] via four wave mixing (FWM). Recently, wideband-tunable THz generation is achieved employing graphene-based nano-antenna enhanced photomixer [20, 21]. The unique challenge in the THz nonlinear generation is the extreme wavelength differences between the optical and the THz waves. The waveguide optimized for optical wave guiding provides weak confinement at the THz regime. So all of previous structures suffer from the limited spatial modal overlap. The metallic slot waveguide provides better overlap than the dielectric waveguides, but the high Ohm loss of metal limited the conversion efficiency [17]. What’s more, the DFG approaches in 2D waveguides could not handle the ultra-high effective refractive of the GSP modes due to the limited dispersion between two optical modes with small frequency differences. In the FWM approaches, the generation in fiber is inefficient due to the weak nonlinear refractive index of silica and multimode propagation of THz waves [18]. To confine the THz-wave, the cross section of section of silicon membrane waveguide is up to several hundred square micrometers. What’s worse, the modal overlap is reduced because the modal profiles concentrate in the air region, which leads to a device length as long as several millimeters [19].

To circumvent the above problems, we propose a direct generation of GSPs via FWM in a compact waveguide incorporating a high-index nano-ridge and two uniformly biased graphene sheets. At both the THz frequencies and the infrared wavelengths, the proposed waveguide provide 2D confinement which significantly reduces the pump powers [22]. To satisfy both the energy conservation and the phase matching condition for the GSP mode featured by ultra-high effective refractive index at the THz regime, here an alternative term of the third order polarization is considered, which corresponds to one pump photon transfers its energy to two signal photons and one GSP photon. To suppress the undesired conversion, the single mode condition for the generated symmetric GSP modes is estimated by the effective index method (EIM). We theoretically investigate the FWM process pumped at 1,550 nm with a peak power of 1 kW by solving the modified coupled mode equations. The generated GSP power reaches its maximum up to 67 W at a propagation distance of only 43.7 μm. The proposed approach have a great potential for integrated chip-scale GSP source.

2. Waveguide concept and nonlinear modeling approach

2.1 waveguide structure

A schematic of the proposed waveguide is shown in Fig. 1 where a silicon ridge is formed between two uniformly biased graphene sheets. The rest regions are filled with KCl [23]. The fabrication is similar to that of the micro-cavity integrated graphene photodetector [24], which is CMOS compatible. To focus on the unique tunability of graphene, the nano-ridge is assumed to have a fixed rectangular shape ( $w = 950 nm, t = 850 nm$ ). Graphene’s complex conductivity is governed by the Kubo Formula, which relates to the radian frequency ω, relaxation time τ, temperature T, and chemical potential μ_c [25]. The value of $τ = μ E_{F} / e V_{F}$ is estimated from the measured DC mobility [26] $μ \approx 10, 000 {cm}^{2} / (Vs)$ for various Fermi levels E_F, where h is the Planck’s constant, e is the charge of an electron and V_F is the Fermi velocity (~10⁸ cm/s in graphene). The chemical potential of graphene can be tuned by the double-gate electrodes [27] or the self-biased schema employing bar-type Au gates [28] as shown in Fig. 1. By modelling graphene as a 1 nm thin layer with effective permittivity [29], the modal properties are investigated at $T = 300 K$ using the commercial finite-element method (FEM) solver COMSOL^TM unless otherwise stated.

Fig. 1 Schematic of the proposed waveguides.

Download Full Size | PDF

2.2 The energy-conservation diagram and the phase-matching condition

The typical degenerate FWM involves two pump photons passing their energy to a signal photon and a THz photon [30]. To cope with the ultra-high effective refractive of the GSP mode, an alternative energy conservation diagram and phase-matching condition [18] are selected as shown in Fig. 2. One pump photon at angular frequency ω₁ ( $λ_{1} = 1550 nm$ ) transfers its energy to two signal photons at ω₂ ( $λ_{1} = 3243.6 nm$ ) and a GSP photon at ω₃ ( $λ_{3} = 35 μm$ ). The relationships illustrated in Fig. 2 can be written as:

ω_{3} = ω_{1} - 2 w_{2}, β_{3} = β_{1} - 2 β_{2} - β_{N L},

where β_NL is the nonlinear phase shift induced by self phase modulation (SPM) and cross phase modulation (XPM). The wave number

β = 2 π Re (N_{e f f}) / λ

is derived from the complex effective index N_eff calculated from COMSOL^TM. Essentially, the schematic in Fig. 2 corresponds to an alternative selection from the 8³ terms of the induced third-order polarization [31]. So the mixing term in conventional coupled mode equations are reformulated as shown in Appendix A for the selected combinations of four waves:

\begin{array}{l} \frac{d A_{1}}{d z} = - \frac{α_{1}}{2} A_{1} + i γ_{11} {| A_{1} |}^{2} A_{1} + 2 i γ_{12} {| A_{2} |}^{2} A_{1} + 2 i γ_{13} {| A_{3} |}^{2} A_{1} + i γ_{1223} A_{2}^{2} A_{3}^{} \exp (i Δ β_{L} z) \\ \frac{d A_{2}}{d z} = - \frac{α_{2}}{2} A_{2} + i γ_{22} {| A_{2} |}^{2} A_{2} + 2 i γ_{21} {| A_{1} |}^{2} A_{2} + 2 i γ_{23} {| A_{3} |}^{2} A_{2} + 2 i γ_{2123} A_{1}^{} A_{2}^{*} A_{3}^{*} \exp (- i Δ β_{L} z) \\ \frac{d A_{3}}{d z} = - \frac{α_{2}}{2} A_{2} + i γ_{33} {| A_{3} |}^{2} A_{3} + 2 i γ_{31} {| A_{1} |}^{2} A_{3} + 2 i γ_{32} {| A_{2} |}^{2} A_{3} + i γ_{3122} A_{1}^{} A_{2}^{* 2} \exp (- i Δ β_{L} z), \end{array}

where A is the slowly varying complex mode amplitude, the attenuation coefficient is defined as

α = 4 π Im (N_{e f f}) / λ

and the nonlinearity coefficient γ is calculated according to [30] with the Kerr index

n^{2} = 5 \times 10^{- 18} m^{2} /W

[32] according to Eq. (15). The linear phase mismatch due to dispersion is defined as

Δ β_{L} = 2 β_{2} + β_{3} - β_{1} .

When the linear phase mismatch Δβ_L is compensated by the nonlinear phase mismatch Δβ_NL, the phase matching is achieved to overcome the dispersive effects and to maximum the conversion rate. Considering a pump with the peak power of 1 kW, the nonlinear phase mismatch is about

β_{N L} \approx (- γ_{11} + 4 γ_{21} + 2 γ_{31}) {| A_{1} |}^{2} \approx 3 \times 10^{5} rad/m

according to Eq. (27) derived in Appendix B. So the desired effective index of GSP mode is 15.05. The phase match was achieved by tuning the chemical potential of graphene sheets when

μ_{c} = 0.909 eV

as depicted in Fig. 3.

Fig. 2 Schematic of (a) the energy conservation diagram and (b) the phase matching condition.

Download Full Size | PDF

Fig. 3 The effective index versus the chemical potential.

Download Full Size | PDF

At THz regime, two parallel graphene sheets support 1D GSP modes [33]. The proposed waveguide supports the 2D GSP mode at the THz frequencies due to the effective index mismatch of 1D GSP caused by the high-index ridge [34, 35]. Since the material loss of graphene is remarkably small compared with noble metal, the graphene based structures outperform the conventional metal based plasmonic waveguides at the modal losses in THz regime [36]. The complex effective index of the GSP mode at the phase match point is 15.05-0.84i. While at the infrared regime, the proposed waveguide is similar to the graphene modulator [37] and the graphene sheets acts as tunable absorption layers. Thus the proposed structure supports low loss photonic modes at the infrared wavelengths due to the Pauli blocking when the chemical potential is higher than half the photon energy [38]. The real part of the effective index of the photonic mode is barely affected by state of the graphene layers [37]. The complex effective index of the optical modes at the phase match point for ω₁ and ω₂, are 3.29-7.27 × 10⁻⁶i and 2.67-1.17 × 10⁻⁴i, respectively.

3. Four wave mixing process

3.1 Modal profiles and single mode propagation

The conversion to the anti-symmetric GSP modes are eliminated because their spatial integrations with the symmetric distributed photonic modes are canceled to zero. To suppress the undesired conversion, the waveguide should be designed to satisfy the single-mode condition for the generated symmetric GSP mode. The key assumption made in the effective index method (EIM) is that most of the power is centralized in the core of the waveguide [39]. The Figs. 4(a)-4(b) shows that most of the power is concentrated in the ridge so that the field magnitudes in the four corners of the waveguide are small enough to be neglected. So the EIM can be employed to estimate the single mode width of generated GSP mode [35]. As illustrated in Fig. 5(a), the proposed waveguide is decomposed into three regions by the dash lines in the first step of EIM. The effective index N_core and N_side of 1D symmetric GSP mode in core region and side regions are then calculated analytically [33] or numerically [40]. The finite width of the ridge is taken into account at the second step of EIM. The proposed waveguide can be evaluated as a trilayer 1D slab waveguide shown in Fig. 5(b). Thus, the cutoff width of the TM₀₁ mode can be calculated as follows [41]:

Fig. 4 (a) The power flow distribution in z direction and (b) the E_y profiles of fundamental symmetric GSP mode at the wavelength of 35 μm; the E_y profiles of (c) the fundamental TM mode at 1,550 nm, (d) the fundamental TM mode at 3,243.6 nm at the phase match point.

Download Full Size | PDF

Fig. 5 Schematic for the (a) first and (b) second step of the EIM

Download Full Size | PDF

w_{th} = \frac{λ_{3}}{2 \sqrt{Re {(N_{c o r e})}^{2} - Re {(N_{s i d e})}^{2}}} .

At the phase match point, the $Re (N_{c o r e})$ and $Re (N_{s i d e})$ for $t = 850 nm$ are 18.82 and 5.87, so the ridge with width less than 979 nm supports single mode propagation of generated symmetric GSP modes. The single mode propagation of generated symmetric GSP mode with $w = 950 nm$ and $t = 850 nm$ at the phase match point is confirmed by COMSOL^TM.

Based on the optical coupling of GSPs [33], the THz mode is confined far below the diffraction limit. As depicted in Figs. 4(b)-4(d), the dielectric ridge of the proposed waveguide provides tight confinement for both the THz wave and the optical waves with a similar concentration at nanoscale. As a result, the modal overlap in the nonlinear integral region (silicon ridge) of optical and THz waves is greatly improved. As listed in Table 1, the nonlinearity coefficients γ for the mixing terms show three orders of magnitude improvement than those of the silicon membrane waveguide [19], which results in the enhanced conversion efficiency.

Table 1. The Nonlinearity Coefficients at Phase Match Point

View Table

3.2 Conversion process

The Fig. 6 shows the FWM process along a propagation distance of 100 μm by solving Eq. (2). For comparison, the peak pump power is 1 kW and the signal peak power is half of the pump peak power [19]. The pump decreases monotonously due to the nonlinear conversion and propagation loss, while the generated GSP power ramps up quickly due to the efficiently energy feeding from pump through FWM. The GSP power reaches its maximum up to 67 W at a propagation distance of only 43.7 μm. The conversion efficiency is $η = 6.7 %$ , which is defined as the ratio of the maximum of generated GSPs power with respect to the input pump power. The conversion efficiency is almost 7 times larger than that of the silicon membrane waveguide with a device length of 6 mm [19].

Fig. 6 The peak power of the pump and generated GSP wave versus the propagation distance

Download Full Size | PDF

The design flow for a certain geometric parameter is summarized in three steps. Firstly, the phase match condition should be derived for certain input photonic modes and satisfied by tuning the graphene sheets. Secondly, the single-mode propagation of generated symmetric GSP mode should be guaranteed. At last, the nonlinearity coefficients is calculated and the coupled mode equations for FWM is solved. A parameter sweep that iterates the design flow can optimize the geometric parameters for a dedicated FWM application.

4. Conclusions

A theoretical study of GSPs generation via FWM using infrared pump waves in a compact waveguide incorporating a silicon nano-ridge and coupled graphene sheets is presented. The proposed waveguide supports GSP modes at the THz frequencies and photonic modes at the infrared wavelengths. The phase match condition can be satisfied by tuning the chemical potential of graphene sheets. To suppress the undesired conversion, the single mode condition for the generated symmetric GSP modes is revealed by the EIM. The degenerated FWM pumped at 1,550 nm with a peak power of 1 kW is theoretically investigated by solving the modified coupled mode equations. The generated GSP power reaches its maximum up to 67 W at a propagation distance of only 43.7 μm. The proposed structure has a great potential for integrated chip-scale GSP source.

Appendix

A. Derivation of the modified coupled mode equations for FWM

The wave equation in a nonlinear medium is:

\nabla^{2} \bar{E} (r, t) - \frac{1}{c^{2}} \frac{\partial \bar{E} {(r, t)}^{2}}{\partial t^{2}} = μ_{0} \frac{\partial \bar{P} {(r, t)}^{2}}{\partial t^{2}} .

The polarization density is a sum of linear and the third order nonlinear parts:

\bar{P} (r, t) = ε_{0} χ \bar{E} (r, t) + \frac{ε_{0} χ_{1111}^{^{(3)}}}{4} \bar{E^{3}} (r, t) = {\bar{P}}_{L} (r, t) + {\bar{P}}_{N L} (r, t) .

The Eq. (4) can be written as:

\nabla^{2} \bar{E} (r, t) - \frac{n^{2}}{c^{2}} \frac{\partial \bar{E} {(r, t)}^{2}}{\partial t^{2}} = μ_{0} \frac{\partial \bar{P_{N L}} {(r, t)}^{2}}{\partial t^{2}} .

Where

n^{2} = 1 + χ

and n is the refractive index. The right hand side of Eq. (6) can be regarded as a source that radiates in a linear medium of refractive index n. Then we define:

S = - μ_{0} \frac{\partial \bar{P_{N L}} {(r, t)}^{2}}{\partial t^{2}} .

Considering a harmonic electric field of angular frequency ω and a complex amplitude E(ω):

\bar{E} (r, t) = \frac{1}{2} [E (ω) \exp (j ω t) + E^{*} (ω) \exp (- j ω t)] .

The Eq. (6) can be written as:

(\nabla^{2} + β^{2}) E (ω) = - S,

where

β = n ω / c .

The source of radiation of the third order nonlinear process is:

S = \frac{1}{8} μ_{0} ε_{0} χ_{1111}^{^{_{(3)}}} {\sum_{l, m, n = \pm 1, \pm 2, \pm 3, \pm 4} (ω_{l} + ω_{m} + ω_{n})}^{2} E_{l} E_{m} E_{n} \exp [j (ω_{l} + ω_{m} + ω_{n}) t]

Considering the degenerated third order nonlinear process

ω_{1} = 2 ω_{2} + ω_{3}

shown in Fig. 2:

\begin{array}{l} S_{1} = \frac{1}{4} μ_{0} ε_{0} ω_{1}^{2} χ_{1111}^{(3)} {3 E_{2}^{2} E_{3} + 3 E_{1} [{| E_{1} |}^{2} + 2 {| E_{2} |}^{2} + 2 {| E_{3} |}^{2}]} \\ S_{2} = \frac{1}{4} μ_{0} ε_{0} ω_{2}^{2} χ_{1111}^{(3)} {6 E_{1} E_{2}^{*} E_{3}^{*} + 3 E_{2} [{| E_{2} |}^{2} + 2 {| E_{1} |}^{2} + 2 {| E_{3} |}^{2}]} \\ S_{3} = \frac{1}{4} μ_{0} ε_{0} ω_{3}^{2} χ_{1111}^{(3)} {3 E_{1} E_{2}^{* 2} + 3 E_{3} [{| E_{3} |}^{2} + 2 {| E_{1} |}^{2} + 2 {| E_{2} |}^{2}]} \end{array}

Under the slow varying envelop approximation, we get:

E (r, ω) = A (z) F (x, y) \exp (j β z) .

The slowly varying complex mode amplitude A is normalized so that

{| A |}^{2}

represents the unit optical power. The left hand side of Eq. (9) can be written as:

(\nabla^{2} + β^{2}) A (z) \exp (j β z) \approx \frac{d A}{d z} j 2 β \exp (j β z) .

Then we get the modified coupled mode equations from Eq. (13) and Eq. (11):

\begin{array}{l} \frac{d A_{1}}{d z} = i γ_{11} {| A_{1} |}^{2} A_{1} + 2 i γ_{12} {| A_{2} |}^{2} A_{1} + 2 i γ_{13} {| A_{3} |}^{2} A_{1} + i γ_{1223} A_{2}^{2} A_{3}^{} \exp (i Δ β_{L} z) \\ \frac{d A_{2}}{d z} = i γ_{22} {| A_{2} |}^{2} A_{2} + 2 i γ_{21} {| A_{1} |}^{2} A_{2} + 2 i γ_{23} {| A_{3} |}^{2} A_{2} + 2 i γ_{2123} A_{1}^{} A_{2}^{*} A_{3}^{*} \exp (- i Δ β_{L} z) \\ \frac{d A_{3}}{d z} = i γ_{33} {| A_{3} |}^{2} A_{3} + 2 i γ_{31} {| A_{1} |}^{2} A_{3} + 2 i γ_{32} {| A_{2} |}^{2} A_{3} + i γ_{3122} A_{1}^{} A_{2}^{* 2} \exp (- i Δ β_{L} z) \end{array}

The nonlinearity coefficient of the waveguide is [30]:

γ = \frac{ω n_{2}}{c} f .

The nonlinear index coefficient is [30]:

n_{2} = \frac{3}{8 n} χ_{1111}^{(3)} .

The overlap integral for SPM and XPM terms are [30]:

f_{l m} = \frac{\iint_{- \infty}^{+ \infty} {| F_{l} |}^{2} {| F_{m} |}^{2} d x d y^{}}{(\iint_{- \infty}^{+ \infty} {| F_{l} |}^{2} d x d y) (\iint_{- \infty}^{+ \infty} {| F_{m} |}^{2} d x d y)}, l, m = 1, 2, 3.

The overlap integral for the mixing terms are [30]:

f_{1223}^{} = \frac{(\iint_{- \infty}^{+ \infty} F_{1}^{*} F_{2} F_{2} F_{3} d x d y)}{\prod_{i = 1, 2, 2, 3} [\iint_{- \infty}^{+ \infty} {| F_{i} |}^{2} d x d y)]^{1 / 2}},

f_{2123}^{} = \frac{(\iint_{- \infty}^{+ \infty} F_{2}^{*} F_{1} F_{2}^{*} F_{3}^{*} d x d y)}{\prod_{i = 2, 1, 2, 3} [\iint_{- \infty}^{+ \infty} {| F_{i} |}^{2} d x d y)]^{1 / 2}},

f_{3122}^{} = \frac{(\iint_{- \infty}^{+ \infty} F_{3}^{*} F_{1} F_{2}^{*} F_{2}^{*} d x d y)}{\prod_{i = 3, 1, 2, 1} [\iint_{- \infty}^{+ \infty} {| F_{i} |}^{2} d x d y)]^{1 / 2}} .

Taking the loss into consideration [42], we then get Eq. (2). The nonlinearity coefficients derived from Eqs. (15-20) for the proposed waveguide at the phase match point are shown in Table 1.

B. Derivation of the phase mismatch condition

We substitute $A = a \exp (j ϕ) = \sqrt{P} \exp (j ϕ)$ into Eq. (2):

\begin{array}{l} exp (i ϕ_{1}) \frac{d a_{1}}{d z} + i a_{1} \exp (i ϕ_{1}) \frac{d ϕ_{1}}{d z} = i a_{1} \exp (i ϕ_{1}) (γ_{11} {| a_{1} |}^{2} + 2 i γ_{12} {| a_{2} |}^{2} + 2 i γ_{13} {| a_{3} |}^{2}) \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} + i γ_{1223} a_{2}^{2} a_{3}^{} \exp [i (Δ β_{L} z + 2 ϕ_{2} + ϕ_{3})] \\ exp (i ϕ_{2}) \frac{d a_{2}}{d z} + i a_{2} exp (i ϕ_{2}) \frac{d ϕ_{2}}{d z} = i a_{2} \exp (i ϕ_{2}) (γ_{22} {| a_{2} |}^{2} a_{2} + 2 i γ_{21} {| a_{1} |}^{2} + 2 i γ_{23} {| a_{3} |}^{2}) \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} + 2 i γ_{2123} a_{1}^{} a_{2}^{} a_{3}^{} \exp [- i (Δ β_{L} z - ϕ_{1} + ϕ_{2} + ϕ_{3})] \\ exp (i ϕ_{3}) \frac{d a_{3}}{d z} + i a_{3} exp (i ϕ_{3}) \frac{d ϕ_{3}}{d z} = i a_{3} \exp (i ϕ_{3}) (γ_{33} {| a_{3} |}^{2} + 2 i γ_{31} {| a_{1} |}^{2} + 2 i γ_{32} {| a_{2} |}^{2}) \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} + i γ_{3122} a_{1}^{} a_{2}^{2} \exp [- i (Δ β_{L} z - ϕ_{1} + 2 ϕ_{2})] \end{array}

Then we divide both side of Eq. (21) by

\exp (i ϕ_{i})

and the multiply both side of Eq. (21) by

2 a_{i} :

\begin{array}{l} \frac{d P_{1}}{d z} + i 2 P_{1} \frac{d ϕ_{1}}{d z} = 2 i [γ_{11} P_{1}^{2} + 2 γ_{12} P_{1} P_{2} + 2 γ_{13} P_{1} P_{3} + γ_{1223} \sqrt{P_{1} P_{2}^{2} P_{3}} \exp (i θ)] \\ \frac{d P_{2}}{d z} + i 2 P_{2} \frac{d ϕ_{2}}{d z} = 2 i [γ_{22} P_{2}^{2} + 2 γ_{21} P_{2} P_{1} + 2 γ_{23} P_{2} P_{3} + 2 γ_{2123} \sqrt{P_{1} P_{2}^{2} P_{3}} \exp (- i θ)] \\ \frac{d P_{3}}{d z} + i 2 P_{3} \frac{d ϕ_{3}}{d z} = 2 i [γ_{33} P_{3}^{2} + 2 γ_{31} P_{3} P_{1} + 2 γ_{32} P_{3} P_{2} + γ_{3122} \sqrt{P_{1} P_{2}^{2} P_{3}} \exp (- i θ)] \end{array}

The relative phase difference between the four involved light waves is defined as:

θ = Δ β z + 2 ϕ_{2} + ϕ_{3} - ϕ_{1} .

Then we employ the Euler's formula

\exp (i θ) = \cos θ + i \sin θ

to separate the real and imaginary part of Eq. (22). The imaginary part is:

\begin{array}{l} \frac{d ϕ_{1}}{d z} = γ_{11} P_{1}^{} + 2 γ_{12} P_{2} + 2 γ_{13} P_{3} + γ_{1223} \sqrt{P_{1}^{- 1} P_{2}^{2} P_{3}} \cos θ \\ \frac{d ϕ_{2}}{d z} = γ_{22} P_{2}^{} + 2 γ_{21} P_{1} + 2 γ_{23} P_{3} + 2 γ_{2123} \sqrt{P_{1} P_{3}} \cos θ \\ \frac{d ϕ_{3}}{d z} = γ_{33} P_{3}^{} + 2 γ_{31} P_{1} + 2 γ_{32} P_{2} + γ_{3122} \sqrt{P_{1} P_{2}^{2} P_{3}^{- 1}} \cos θ \end{array}

So we get:

\begin{array}{l} \frac{d θ}{d z} = Δ β + 2 \frac{d φ_{2}}{d z} + \frac{d φ_{3}}{d z} - \frac{d φ_{1}}{d z} = Δ β_{L} + (2 γ_{22} + 2 γ_{32} - 2 γ_{12}) P_{2} + (- γ_{11} + 4 γ_{21} + 2 γ_{31}) P_{1} \\ \begin{matrix}  \end{matrix} + (4 γ_{23} - 2 γ_{13} + γ_{33}) P_{3} + [4 γ_{2123} \sqrt{P_{1} P_{3}} + γ_{1223} \sqrt{P_{1}^{- 1} P_{2}^{2} P_{3}} - γ_{3122} \sqrt{P_{1} P_{2}^{2} P_{3}^{- 1}}] \cos θ \end{array}

Operating in a phase match condition remains near

θ = π / 2

[43], the last term may be neglected:

\frac{d θ}{d z} \approx Δ β {}_{L}+ (- γ_{11} + 4 γ_{21} + 2 γ_{31}) P_{1} .

When the linear phase mismatch Δβ_L is compensated by the nonlinear phase mismatch β_NL, i.e.:

- Δ β {}_{L}= β {}_{N L}\equiv (- γ_{11} + 4 γ_{21} + 2 γ_{31}) P_{1} .

The phase matching is achieved to overcome the dispersive effects and to maximum the conversion rate.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities with Grant No. BLX2014-26 and NSFC with Grant No. 61300180 and 51378156.

References and links

1. A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]

2. C. Yen and A. Alu, “Terahertz Metamaterial Devices Based on Graphene Nanostructures,” IEEE Trans. Terahertz Sci. Technol. 3(6), 748–756 (2013). [CrossRef]

3. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009). [CrossRef]

4. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef] [PubMed]

5. A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]

6. P. Avouris and M. Freitag, “Graphene Photonics, Plasmonics, and Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20(1), 6000112 (2014). [CrossRef]

7. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012). [PubMed]

8. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012). [PubMed]

9. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013). [CrossRef]

10. J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013). [CrossRef] [PubMed]

11. M. Esquius-Morote, J. S. Gomez-Diaz, and J. Perruisseau-Carrier, “Sinusoidally Modulated Graphene Leaky-Wave Antenna for Electronic Beamscanning at THz,” IEEE Trans. Terahertz Sci. Technol. 4(1), 116–122 (2014). [CrossRef]

12. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of Plasmonic Waves in Graphene by Guided-Mode Resonances,” ACS Nano 6(9), 7806–7813 (2012). [CrossRef] [PubMed]

13. J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013). [CrossRef] [PubMed]

14. M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013). [CrossRef] [PubMed]

15. C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express 16(17), 13296–13303 (2008). [CrossRef] [PubMed]

16. Y. Takushima, S. Y. Shin, and Y. C. Chung, “Design of a LiNbO3 ribbon waveguide for efficient difference-frequency generation of terahertz wave in the collinear configuration,” Opt. Express 15(22), 14783–14792 (2007). [CrossRef] [PubMed]

17. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17(16), 13502–13515 (2009). [CrossRef] [PubMed]

18. K. Suizu and K. Kawase, “Terahertz-wave generation in a conventional optical fiber,” Opt. Lett. 32(20), 2990–2992 (2007). [CrossRef] [PubMed]

19. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012). [CrossRef] [PubMed]

20. P. Chen and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photonics 1(8), 647–654 (2014). [CrossRef]

21. M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable terahertz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101(21), 214102 (2012). [CrossRef]

22. S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004). [CrossRef]

23. Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013). [CrossRef]

24. M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012). [CrossRef] [PubMed]

25. L. Falkovsky and A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007). [CrossRef]

26. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films,” Science 306(5696), 666–669 (2004). [CrossRef] [PubMed]

27. C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013). [CrossRef]

28. J. Gomez-Diaz, C. Moldovan, S. Capdevilla, J. Romeu, L. Bernard, A. Magrez, A. Ionescu, and J. Perruisseau-Carrier, “Self-biased Reconfigurable Graphene Stacks for Terahertz Plasmonics,” arXiv preprint arXiv:1405.3320 (2014).

29. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

30. G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000).

31. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

32. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003). [CrossRef]

33. B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012). [CrossRef]

34. M. Liscidini, “Surface guided modes in photonic crystal ridges: the good, the bad, and the ugly,” J. Opt. Soc. Am. B 29(8), 2103–2109 (2012). [CrossRef]

35. Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Opt. Commun. 328, 124–128 (2014). [CrossRef]

36. B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013). [CrossRef] [PubMed]

37. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011). [CrossRef] [PubMed]

38. Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009). [CrossRef]

39. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

40. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011). [CrossRef]

41. H. Kogelnik and V. Ramaswamy, “Scaling Rules for Thin-Film Optical Waveguides,” Appl. Opt. 13(8), 1857–1862 (1974). [CrossRef] [PubMed]

42. J. Zhang, E. Cassan, D. Gao, and X. Zhang, “Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics,” Opt. Express 21(12), 14876–14887 (2013). [CrossRef] [PubMed]

43. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002). [CrossRef]

Nonlinearity Coefficients	Value (1/W/m)
γ₁₁	1.515 × 10²
γ₂₂	6.192 × 10¹
γ₃₃	5.557 × 10⁰
γ₁₂, γ₂₁	1.031 × 10²
γ₂₃, γ₃₂	4.938 × 10¹
γ₁₃, γ₃₁	3.522 × 10¹
γ₁₂₂₃	4.552 × 10¹
γ₂₁₂₃	4.552 × 10¹
γ₃₁₂₂	4.411 × 10¹

Direct generation of graphene plasmonic polaritons at THz frequencies via four wave mixing in the hybrid graphene sheets waveguides

Abstract

1. Introduction

2. Waveguide concept and nonlinear modeling approach

2.1 waveguide structure

2.2 The energy-conservation diagram and the phase-matching condition

3. Four wave mixing process

3.1 Modal profiles and single mode propagation

3.2 Conversion process

4. Conclusions

Appendix

A. Derivation of the modified coupled mode equations for FWM

B. Derivation of the phase mismatch condition

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (27)

Optics Express