Optimized nonlinear terahertz response of graphene in a parallel-plate waveguide

Navaeipour, Parvin; Dignam, Marc M.

doi:10.1063/1.5045652

Third harmonic generation of terahertz radiation is expected to occur in monolayer graphene due to the nonlinear relationship between the crystal momentum and the current density. In this work, we calculate the terahertz nonlinear response of monolayer graphene inside a parallel-plate waveguide including pump depletion, self-phase, and cross-phase modulation. To overcome the phase mismatching between the pump field and third-harmonic field at high input fields due to self-phase and cross-phase modulation, we design a waveguide with two dielectric layers with different indices of refraction. We find that, by tuning the relative thicknesses of the two layers, we are able to improve phase matching and thereby increase the power efficiency of the system by more than a factor of two at high powers. With this approach, we find that despite the loss in this system, for an incident frequency of 2 THz, we are able to achieve power efficiencies of 75% for graphene with low Fermi energies of 20 meV and up to 35% when the Fermi energy is 100 meV.

I. INTRODUCTION

Graphene, as a zero-bandgap two-dimensional semiconductor with a linear electron band dispersion near the Dirac points, has the potential to exhibit very interesting nonlinear optical properties.^1–4 The linear dispersion relation of the electrons near the Dirac points leads to a constant electron speed.^5–7 Thus, the intraband current induced in graphene by terahertz (THz) fields displays clipping as the amplitude of the incident field increases, which generates odd harmonics in the current and transmitted electric field.^8–12 Exploiting the nonlinear response of graphene enables one to produce higher frequency THz radiation through the generation of harmonics. Several experimental and theoretical groups have examined third-harmonic generation from graphene at terahertz frequencies. Almost all have employed a configuration where the field is normally incident on the graphene.^13–15 The third harmonic field generated in those experiments is expected to be very small and there is only one paper¹⁶ that has detected the third harmonic in monolayer graphene at THz frequencies. Therefore, it is important to find a way to enhance the generation.

In this work, we consider a configuration where the radiation propagates in a metallic parallel-plate waveguide (PPW), with the monolayer graphene sheet lying at the midpoint between the two plates as shown in Fig. 1.¹⁷ With this configuration, we increase the interaction time between the radiation and graphene and thereby generate a larger-amplitude harmonic field. We have shown in previous work that this configuration can increase the power efficiency of the system by more than a factor of 100, relative to the results for the normal-incidence configuration, and that the power efficiency is relatively insensitive to the plate separation but depends strongly on the Fermi energy.¹⁷ However, in Ref. 17, we did not include the effects of pump depletion, self-phase modulation (SFM), and cross-phase modulation (XFM) in our calculations. In this work, we develop a coupled-mode theory including all propagating lossy modes to calculate the power efficiency for third-harmonic generation in a PPW and use this model to examine the impact of these effects on the power conversion efficiency.¹⁸

FIG. 1.

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The metallic parallel-plate waveguide with monolayer graphene inside forms the system being modelled. The inner material of the waveguide is polyolefin, and the graphene is placed at the center of the waveguide at y = b/2. The pump field propagates in the +z-direction and is polarized in the x-direction.

For weak input fields, there is generally good phase matching in the waveguide between the pump field in the TE₁ mode at ω and the third harmonic field in the TE₃ mode at 3ω. However, as shown later, when the pump field amplitude increases, the phase matching degrades due to SFM and XFM. To overcome this, we propose a new configuration in which the waveguide contains two layers of dielectric materials: cyclic polyolefin (n₁ = 1.53) and phenol formaldehyde resin (n₂ = 1.70).²⁰ One goal in this work is to optimize the thickness of the dielectric layers and the Fermi energy of the graphene to obtain phase matching and thereby maximize the generated third-harmonic electric field.

The paper is organized as follows: In Sec. II, we expand the electric field at the fundamental and third harmonic in terms of the lossy modes of the PPW and use the slowly varying envelope approximation to derive the differential equations for the mode amplitudes. In Sec. III, we compare the results obtained for the generated third-harmonic field, using our coupled-mode theory in the undepleted pump approximation, calculation with pump depletion, and using a full calculation, which includes pump depletion and self- and cross-phase modulation. In Sec. IV, we propose a new configuration PPW and demonstrate that this configuration allows us to essentially eliminate phase mismatch over a wide range of Fermi energies and input field amplitudes. Finally, in Sec. V, we summarize our results.

II. THEORY

In this section, we first solve for the lossy linear modes of the waveguide in the presence of graphene. We then expand the fields at ω and 3ω in terms of these linear modes, to derive the nonlinear coupled mode equations.

Our parallel-plate waveguide consists of two metallic plates placed at y = 0 and y = b, with the monolayer graphene midway between the plates at y = b/2 as shown in Fig. 1. The inner material of the waveguide is chosen to be cyclic polyolefin, with a refractive index of n₁ = 1.53, due to its compatibility with graphene, ease of fabrication, and low loss at THz frequencies.²⁰ The THz wave propagates in the +z direction and is polarized in the x direction. For simplicity, we take the plates to be perfect conductors that are infinite in the x direction.

From Maxwell’s equations, we obtain the inhomogeneous wave equation¹⁹

\nabla \times \nabla \times E (r, t) = μ_{0} ϵ_{0} \frac{\partial^{2} E (r, t)}{\partial t^{2}} + μ_{0} \frac{\partial J (r, t)}{\partial t},

(1)

where $E (r, t)$ is the total electric field and $J (r, t)$ is the total current density in the graphene. The incident electric field is taken to be harmonic with frequency of ω and a third-harmonic electric field is generated so that the total electric field is

E (r, t) = E (r; ω) e^{- i ω t} + E (r; 3 ω) e^{- i 3 ω t} + c . c .

(2)

and the total current density is

J (r, t) = J (r; ω) e^{- i ω t} + J (r; 3 ω) e^{- i 3 ω t} + c . c .

(3)

Now, the current density can be broken up into its linear and nonlinear components as J = J_L + J_NL. Here, J_L is the current due to the linear conductivity of the graphene and is given by

J_{L} (r; ω^{'}) = σ^{(1)} (ω^{'}) E (r; ω^{'}) δ (y - b / 2),

(4)

where σ⁽¹⁾(ω′) is the linear conductivity of the graphene, with ω′ = {ω, 3ω}. The linear conductivity has both intraband and interband contributions.^9,17,23 At low Fermi energies (E_F ≤ 20 meV), a photon with frequency of f₀ ≡ ω₀/2π = 2 THz is not able to cause interband transitions. Thus, the dominant contribution to the linear conductivity is the intraband conductivity at ω₀.¹⁷ However, for the third harmonic, the interband transition does contribute to some degree in the linear conductivity at low Fermi energies and so is included in our calculations.¹⁷ The intraband contribution to the linear conductivity of monolayer graphene at zero temperature is proportional to the Fermi energy and is given by²³

σ^{(1)} (ω) = \frac{e^{2} τ E_{F}}{π ћ^{2} (1 + i ω τ)},

(5)

where E_F is the Fermi energy and τ is the phenomenological scattering time, which in this work is taken to be 50 fs. Thus, to limit linear loss, it is better to work at low Fermi energies.

The nonlinear current density of the graphene, J_NL, arises from the third-order nonlinear conductivity, σ⁽³⁾. At 3ω, it is given by

\begin{matrix} J_{N L} (r; 3 ω) & = & μ_{0} σ^{(3)} (3 ω; ω, ω, ω) \{) E (r; ω) {\})}^{3} δ (y - b / 2) \\ + 3 μ_{0} σ^{(3)} (3 ω; 3 ω, - 3 ω, 3 ω) E (r; 3 ω) \\ \times | E (r; 3 ω) |^{2} δ (y - b / 2) + 6 μ_{0} σ^{(3)} (3 ω; 3 ω, - ω, ω) \\ \times E (r; 3 ω) | E (r; ω) |^{2} δ (y - b / 2) . \end{matrix}

(6)

The first term in Eq. (6) is the most important, as it is the source of the third-harmonic electric field. The next two terms are related to SFM and XFM, respectively.

The nonlinear current density at ω is given by

\begin{matrix} J_{N L} (r; ω) & = & 3 μ_{0} σ^{(3)} (ω; 3 ω, - ω, - ω) E (r; 3 ω) \{) E^{*} (r; ω) {\})}^{2} δ (y - b / 2) \\ + 3 μ_{0} σ^{(3)} (ω; ω, - ω, ω) E (r; ω) | E (r; ω) |^{2} δ (y - b / 2) \\ + 6 μ_{0} σ^{(3)} (ω; ω, - 3 ω, 3 ω) E (r; ω) | E (r; 3 ω) |^{2} δ (y - b / 2) . \end{matrix}

(7)

The first term in Eq. (7) gives the nonlinear current at the graphene that results in pump depletion and the next two terms represent SPM and XFM, respectively. The factors of 3 and 6 in Eqs. (6) and (7) arise from the number of ways of generating the nonlinear polarization in these cases.

In Eqs. (6) and (7), σ⁽³⁾ is the nonlinear conductivity of monolayer graphene. In this work, we use the theoretical expression of Cheng et al.,²² which is derived from perturbative calculations at zero temperature, for electrons close to the Dirac points, with the neglect of the effects of the scattering (τ → ∞). Under these assumptions,

σ^{(3)} (3 ω; ω, ω, ω) = \frac{i σ_{0} {({ћ v}_{F} e)}^{2}}{48 π {(ћ ω)}^{4}} T (\frac{ћ ω}{2 E_{F}}),

(8)

where $σ_{0} = e^{2} / 4 ћ$ is the universal conductivity of the graphene, $v$ _F is the Fermi velocity of the electrons in the graphene, taken to be 1.1 × 10⁶ m/s, and

T (x) \equiv 17 G (x) - 64 G (2 x) + 45 G (3 x),

(9)

in which

G (x) \equiv \ln (\frac{1 + x}{1 - x}) + i π Θ (| x | - 1),

(10)

where Θ(x) is the Heaviside step function. The other nonlinear conductivities in Eqs. (6) and (7) are related to σ⁽³⁾(3ω; ω, ω, ω) by¹⁹

\begin{matrix} σ^{(3)} (3 ω; ω, ω, ω) & = & σ^{(3)} (3 ω; 3 ω, - 3 ω, 3 ω) = σ^{(3)} (3 ω; 3 ω, - ω, ω) \\ = & 3 σ^{(3)} (ω; 3 ω, - ω, - ω) = 3 σ^{(3)} (ω; ω, - ω, ω) \\ = & 3 σ^{(3)} (ω; ω, - 3 ω, 3 ω) . \end{matrix}

(11)

A. Linear modes

For harmonic waves travelling in the +z direction with angular frequency ω, the linear electric field for the nth transverse electric (TE) mode is given by^17,21

E_{x}^{(n)} (y, z; ω) = \{) \begin{matrix} E_{n} e^{i {\tilde{β}}_{n} (ω) z} \sin [{\tilde{k}}_{n} (ω) y] & y > b / 2 \\ - E_{n} e^{i {\tilde{β}}_{n} (ω) z} \sin [{\tilde{k}}_{n} (ω) (y - b / 2)] & y < b / 2 \end{matrix},

(12)

where n = 1, 2, 3, …, E_n is the amplitude of the nth mode, and ${\tilde{k}}_{n}$ is the complex wave number for the field’s y dependence. This wavenumber depends on the linear conductivity of the graphene and is obtained by enforcing the boundary conditions at the graphene, which leads to the following transcendental equation:¹⁷

σ^{(1)} (ω) = \frac{4 i {\tilde{ϕ}}_{n}}{ω μ b} \cot ({\tilde{ϕ}}_{n}),

(13)

where ${\tilde{ϕ}}_{n} \equiv {\tilde{k}}_{n} (ω) b / 2$ and μ is the permeability of the dielectric. The complex propagation constant, ${\tilde{β}}_{n}$ ⁠, of the TE_n mode is given by

{\tilde{β}}_{n} (ω) = \sqrt{{(\frac{n_{1} ω}{c_{0}})}^{2} - {\tilde{k}}_{n}^{2}},

(14)

where c₀ is the speed of light in a vacuum and n₁ is the refractive index of the dielectric material. If there is no graphene, i.e., for a bare waveguide, ${\tilde{k}}_{n} \equiv k_{n}^{0} = n π / b$ ⁠, where n is an integer.

B. Coupled mode equations

In this work, we take the input field at ω to be in the TE₁ mode at z = 0. We expand the field at ω in terms of the lossy TE_n modes as

E (y, z; ω) = \sum_{n} A_{n} (z; ω) e^{i {\tilde{β}}_{n} (ω) z} \sin ({\tilde{k}}_{n} (ω) y)

(15)

and expand the generated third-harmonic electric field as

E (y, z; 3 ω) = \sum_{n} A_{n} (z; 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} (3 ω) y),

(16)

where the summation is over all of the lossy modes propagating in the waveguide and the A_n(z; ω) are slowly varying envelopes. Although this is not an exact expansion (as the lossy modes are not complete), we showed in our previous work¹⁷ that using this expansion in the undepleted pump approximation, we obtain almost identical results for the generated third-harmonic field similar to those obtained using an exact Green function approach as long as the frequency is not close to the cut-off frequency.

The initial conditions are as follows: $A_{n} (0; ω) = δ_{n, 1} E_{i n p u t} / \sin ({\tilde{k}}_{n} (ω) \frac{b}{2})$ and A_n(0; 3ω) = 0 where E_input is the amplitude of the incident field at the graphene. We now employ our mode expansions to solve Eq. (1) for the fundamental and third harmonic fields including pump-depletion, SFM, and XFM. Substituting Eqs. (6), (7), and (14) into Eq. (1) along with the facts that ∇ · E = 0 and the modes are essentially orthogonal gives

\begin{matrix} - \nabla^{2} E (y, z; 3 ω) & = & i 3 ω μ_{0} σ^{(1)} (3 ω) E (y, z; 3 ω) δ (y - b / 2) + 3 i ω μ_{0} σ^{(3)} (3 ω; ω, ω, ω) \{) E (y, z; ω) {\})}^{3} δ (y - b / 2) \\ + 3 i ω μ_{0} σ^{(3)} (3 ω; 3 ω, - 3 ω, 3 ω) E (y, z; 3 ω) | E (y, z; 3 ω) |^{2} δ (y - b / 2) \\ + 6 i ω μ_{0} σ^{(3)} (3 ω; 3 ω, - ω, ω) E (y, z; 3 ω) | E (y, z; ω) |^{2} δ (y - b / 2) + 9 μ_{0} ϵ ω^{2} E (y, z; 3 ω) . \end{matrix}

(17)

Now, using Eqs. (15) and (16) and employing the slowly varying envelope approximation [i.e., neglecting $d^{2} A_{n} (z; 3 ω) / d z^{2}$ ], we obtain the following differential equation for the amplitude of the electric field at 3ω for the mth mode (see the Appendix for details):

\begin{matrix} \frac{d A_{m} (z; 3 ω)}{d z} & = & \frac{- 3 ω μ_{0}}{b {\tilde{β}}_{m} (3 ω)} \{) \sum_{n^{'}} σ^{(3)} (3 ω; ω, ω, ω) \{) A_{n^{'}} (z; ω) {\})}^{3} e^{i (3 {\tilde{β}}_{n^{'}} (ω) - {\tilde{β}}_{m} (3 ω)) z} S_{m}^{* (3)} () \frac{b}{2})) \{) S_{n^{'}}^{(1)} () \frac{b}{2})) {\})}^{3} \\ + \sum_{n} σ^{(3)} (3 ω; 3 ω, - 3 ω, 3 ω) A_{n} (z; 3 ω) | A_{n} (z; 3 ω) |^{2} e^{i ({2 \tilde{β}}_{n} (3 ω) - {\tilde{β}}_{m} (3 ω) - {\tilde{β}}_{n}^{*} (3 ω)) z} S_{m}^{* (3)} () \frac{b}{2})) S_{n}^{* (3)} () \frac{b}{2})) \{) S_{n}^{(3)} () \frac{b}{2})) {\})}^{2} \\ + \sum_{n} \sum_{n^{'}} 2 σ^{(3)} (3 ω; 3 ω, - ω, ω) A_{n} (z; 3 ω) | A_{n^{'}} (z; ω) |^{2} e^{i ({\tilde{β}}_{n} (3 ω) - {\tilde{β}}_{m} (3 ω) + {\tilde{β}}_{n^{'}} (ω) - {\tilde{β}}_{n^{'}}^{*} (ω)) z} S_{m}^{* (3)} () \frac{b}{2})) S_{n^{'}}^{* (1)} () \frac{b}{2})) S_{n^{'}}^{(1)} () \frac{b}{2})) S_{n}^{(3)} () \frac{b}{2})) \}), \end{matrix}

(18)

where $S_{n}^{(1)} () y)) \equiv \sin ({\tilde{k}}_{n} (ω) y)$ and $S_{n}^{(3)} () y)) \equiv \sin ({\tilde{k}}_{n} (3 ω) y)$ ⁠. Similarly, for the electric field modes at ω, we obtain

\begin{matrix} \frac{d A_{m} (z; ω)}{d z} & = & \frac{- 3 ω μ_{0}}{b {\tilde{β}}_{m} (ω)} \sum_{n} \{) \sum_{n^{'}} σ^{(3)} (ω; 3 ω, - ω, - ω) A_{n^{'}} (z; 3 ω) \{) A_{n}^{*} (z; ω) {\})}^{2} e^{i ({\tilde{β}}_{n^{'}} (3 ω) - 2 {\tilde{β}}_{n}^{*} (ω) - {\tilde{β}}_{m} (ω)) z} {S_{m}^{*}}^{(1)} () \frac{b}{2})) S_{n^{'}}^{(3)} () \frac{b}{2})) \{) S_{n}^{* (1)} () \frac{b}{2})) {\})}^{2} \\ + σ^{(3)} (ω; ω, - ω, ω) A_{n} (z; ω) | A_{n} (z; ω) |^{2} e^{i (2 {\tilde{β}}_{n} (ω) - {\tilde{β}}_{n}^{*} (ω) - {\tilde{β}}_{m} (ω)) z} S_{m}^{* (1)} () \frac{b}{2})) \{) S_{n}^{(1)} () \frac{b}{2})) {\})}^{2} S_{n}^{* (1)} () \frac{b}{2})) \\ + \sum_{n^{'}} 2 σ^{(3)} (ω; ω, - 3 ω, 3 ω) A_{n} (z; ω) |) A_{n^{'}} (z; 3 ω) {|)}^{2} e^{i ({\tilde{β}}_{n} (ω) + {\tilde{β}}_{n^{'}} (3 ω) - {\tilde{β}}_{n^{'}}^{*} (3 ω) - {\tilde{β}}_{m} (ω)) z} S_{m}^{* (1)} () \frac{b}{2})) S_{n}^{(1)} () \frac{b}{2})) |) S_{n^{'}}^{(3)} () \frac{b}{2})) {|)}^{2} \}) . \end{matrix}

(19)

III. RESULTS

In this section, we solve the coupled dynamic equations for the amplitudes of the electric fields at 3ω and ω, given by Eqs. (18) and (19), respectively. In all that follows, we take the incident (pump) field to have frequency f₀ ≡ ω₀/2π = 2 THz and take the plate separation to be 70 μm. We choose this plate separation because it is the largest separation for which there are only two propagating modes in the waveguide at 3ω. Note also that for this plate separation, only the TE₁ mode is a propagating mode at ω. In our previous work, we showed that there is a perfect phase matching between the first mode at ω and the third mode at 3ω when there is no graphene. We also showed in that work that the loss decreases, group velocity dispersion decreases, and the phase matching improves as the plate separation increases. However, increasing the plate separation also decreases the amplitude of the incident field at the graphene for a fixed input power, so there is a trade-off. We choose the separation of 70 μm as a compromise and to avoid the potential problems (such as intermode-scattering) that having multiple propagating modes can introduce. Thus, we only include first and third modes in our calculations. To solve the coupled equations (18) and (19), we employ a Runge-Kutta algorithm; solving these coupled equations numerically takes less than 1 min on an i7 processor.

In Fig. 2, we plot the generated third-harmonic electric field at the graphene as a function of z for a Fermi energy of E_F = 50 meV for three different input field amplitudes (E_input = 5, 10, and 15 kV/cm). We present the results of three different calculations: undepleted pump approximation (red solid curves), calculation with pump depletion (green dotted curves), and the full calculation that includes the SFM and XFM and pump depletion (blue dashed curves). To demonstrate the importance of pump depletion, we first present the results of the calculation when the SFM, XFM, and pump depletion are neglected. This is accomplished by keeping only the first term on the right-hand side of Eqs. (18) and (19) and taking A₁(z; ω) = A₁(0; ω). Note, however, that we still include linear loss in all modes. Initially the third-harmonic grows rapidly, while the field at ω (see the insets) decays exponentially until it is essentially gone after a propagation distance of about 2 mm. Oscillations in the third-harmonic arise due to the phase mismatch between the third-harmonic field in the n = 1 and n = 3 modes, which is why they persist even after the fundamental field is essentially gone. Almost identical results are obtained for the generated third-harmonic field at low input fields when pump depletion is included, as shown in Fig. 2(a). However, as we increase the input field, we see a significant reduction in the generated third-harmonic field with pump depletion relative to the undepleted pump approximation. This is due to the increased transfer of power from the incident (pump) field to the third harmonic field. This can also be shown in the insets of Figs. 2(b) and 2(c), for strong input fields, where the pump field decays faster when the pump field is increased.

FIG. 2.

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Generated third-harmonic electric field at the graphene calculated in the undepleted pump approximation (red solid curves), calculated with pump depletion (green dotted curves), and calculated with a full calculation (blue dashed curves) for different input fields of (a) 5 kV/cm, (b) 10 kV/cm, and (c) 15 kV/cm for a Fermi energy of E_F = 50 meV. In the inset, we plot the fundamental field at the graphene as a function of z.

We now turn to the effects of self- and cross-phase modulation. As we increase the input field, in Figs. 2(b) and 2(c), we observe that including SFM and XFM results in a decrease in the generated third-harmonic electric field. This is due to a degradation in the phase matching between the third mode at 3ω and the first mode at ω due to SFM and XFM. For an input field of 15 kV/cm, this results in a 6% reduction in the peak field. As shown later, the effect is much more significant at lower Fermi energies and/or higher input fields. Let us now examine the effects of SFM and XFM on phase matching in the PPW in more detail.

It is easy to show, using Eq. (14), that for a bare waveguide, there is a perfect phase matching between the n = 3 mode at 3ω and the first mode at ω.¹⁷ To generate a strong third-harmonic electric field, we need to have a very small effective refractive index difference between these two modes in the presence of graphene loss and SFM and XFM. This effective refractive index difference for a lossy waveguide with SFM and XFM is approximately given by

Δ n_{e ff} \equiv n_{e ff}^{(3)} (3 ω) - n_{e ff}^{(1)} (ω),

(20)

where

n_{e f f}^{(1)} (ω) \equiv \frac{Re {{\tilde{β}}_{1} (ω)}}{ω / c_{0}} + n_{2}^{(1)} (ω) | E_{i n p u t} |^{2}

(21)

and

n_{e ff}^{(3)} (3 ω) \equiv \frac{Re {{\tilde{β}}_{3} (3 ω)}}{3 ω / c_{0}} + n_{2}^{(3)} (3 ω) | E_{i n p u t} |^{2} .

(22)

The first terms in each of Eqs. (21) and (22) are the input-field-independent effective refractive indices for the first mode at ω and third mode at 3ω, respectively. The second terms are added to approximately account for the change in the effective index due to the SFM and XFM. Using Eqs. (18) and (19), we obtain

\begin{matrix} n_{2}^{(1)} (ω) ≃ & Im \{- \frac{3 μ_{0} c_{0}}{b {\tilde{β}}_{1} (ω)} σ^{(3)} (ω; ω, - ω, ω) | S_{1}^{(1)} () \frac{b}{2})) |^{2}\}, \\ n_{2}^{(3)} (3 ω) ≃ & Im \{- \frac{2 μ_{0} c_{0}}{b {\tilde{β}}_{3} (3 ω)} σ^{(3)} (3 ω; 3 ω, - ω, ω) | S_{3}^{(3)} () \frac{b}{2})) |^{2}\} . \end{matrix}

(23)

In deriving these expressions, we have included only the terms proportional to the square of the electric field at ω and have taken the pump field to be given by its value at z = 0. We find that the terms proportional to the square of the third-harmonic electric field are negligible relative to the linear electric field. However, in our full numerical calculations, all terms are retained, as is the z-dependence of A₁(ω).

The effective index difference between the third mode at 3ω and the first mode at ω as a function of Fermi energy is shown in Fig. 3 for different input pump field amplitudes. When E_input = 0, Δn_eff linearly increases with Fermi energy, due to the dependence of the propagation constants on the doping level of the graphene. As the input field increases, Δn_eff increases for all Fermi energies, but increases the most for low Fermi energies. At the lowest Fermi energy of 20 meV and input field of 15 kV/cm, Δn_eff is very large, reaching a value of approximately 0.06. This strong dependence on Fermi energy has its origin in the strong dependence of the nonlinear conductivity on Fermi energy, as shown in Eqs. (8)–(10).

FIG. 3.

$FIG. 3. Power-dependent effective refractive index difference between the TE3 mode at 3ω and the TE1 mode at ω, as a function of Fermi energy for four different input fields. When Einput = 0, there are no effects due to XFM and SFM present.$

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Power-dependent effective refractive index difference between the TE₃ mode at 3ω and the TE₁ mode at ω, as a function of Fermi energy for four different input fields. When E_input = 0, there are no effects due to XFM and SFM present.

We now consider the power efficiency S_eff of the device, which is defined as the ratio of the power in the third harmonic at the end of the waveguide to the power in the fundamental at the beginning of the waveguide. In Fig. 4, we plot the maximum power efficiency as a function of Fermi energy in three different schemes: undepleted-pump approximation, calculation with pump depletion, and full calculation. In all cases, the length of the waveguide is chosen to be the distance at which the power in the third harmonic field is a maximum, as shown in Fig. 2. Decreasing the Fermi energy leads to higher nonlinear conductivity and lower linear conductivity by the graphene. Therefore, we obtain a higher power efficiency as the Fermi energy is decreased. We see in Fig. 4(a) that for a weak input field of 5 kV/cm, the power efficiency is almost the same in all three calculation schemes. It is only at low Fermi energies (∼20 meV) that there is a noticeable difference in the power efficiency. As we increase the input field, we see in Figs. 4(b) and 4(c) that the effects of pump depletion, SFM, and XFM become very significant, particularly at low Fermi energies. For strong enough fields and low Fermi energies, our calculations for the undepleted pump approximation yield a non-physical power efficiency that is greater than 100%.¹⁷ However, including pump depletion results in power efficiencies less than 100%, as required. It is observed that the power efficiency decreases when SFM and XFM are included. For example, for Fermi energy of E_F = 20 meV, when the input field is E_input = 10 kV/cm, the power efficiency decreases from S_eff = 48.26% in the PPW with pump depletion to S_eff = 28.13% in the PPW with full calculation, while for an input field of E_input = 15 kV/cm, the power efficiency decreases from 64.47% to 34.64%. It is therefore worth examining if we can modify the structure to decrease the phase mismatch introduced by SFM and XFM.

FIG. 4.

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Power efficiency of the waveguide as a function of Fermi energy calculated in the undepleted pump approximation (red solid line), calculated with pump depletion (green dotted line), and for the full calculation (blue dashed line) for different input fields of (a) 5 kV/cm, (b) 10 kV/cm, and (c) 15 kV/cm.

IV. NEW CONFIGURATION: MITIGATING SELF- AND CROSS-PHASE MODULATION

In this section, we define a new configuration of the PPW in order to deal with phase mismatching due to the SFM and XFM. We consider the waveguide shown in Fig. 5, where there are two different dielectric materials in the waveguide: cyclic polyolefin with refractive index of n₁ = 1.53 and phenol-formaldehyde resin with refractive index of n₂ = 1.70. We choose phenol formaldehyde resin as the second dielectric because it has a higher index than cyclic polyolefin, it has low absorption at THz frequencies, and it is compatible with graphene and cyclic polyolefin. The monolayer graphene is located at y = b/2 midway between two metallic plates. The n₁ material is in the regions y = 0 to y = d₁ and y = b − d₁ to y = b, while the n₂ material is in the region y = d₁ to y = b − b₁. As shown later, this new configuration allows us to control to some degree the phase matching between the third mode at 3ω and the first mode at ω. In the following, we optimize d₁ to obtain the best phase matching between the TE₃ mode at 3ω and TE₁ mode at ω and thereby maximize the third harmonic generation and the power efficiency. Note that in all cases, the total plate separation is fixed at b = 70 μm.

FIG. 5.

$FIG. 5. New configuration parallel plate waveguide where we have two different dielectric layers with different indices of refraction. We use this structure to reduce phase mismatch by optimizing the thickness, d1, of the material with the lower index.$

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New configuration parallel plate waveguide where we have two different dielectric layers with different indices of refraction. We use this structure to reduce phase mismatch by optimizing the thickness, d₁, of the material with the lower index.

A. Phase matching

In this section, we examine the effect of the layer thickness on the linear phase mismatch. To improve the phase matching in the waveguide, we need to lower the effective index of refraction for the TE₃ mode relative to that of the TE₁ mode. Thus, we wish to alter the waveguide structure such that the modification affects the effective index for the TE₁ and TE₃ modes differently. At high frequency, the effective index of any mode is given by an average of the indices of refraction of the dielectrics in the waveguide, and the average will roughly depend upon what fraction of the mode energy lies in each of the regions with different indices. In Fig. 6, we plot the normalized TE₁ mode at ω and the TE₃ mode at 3ω for a 2 THz field in a 70 μm waveguide in our original configuration and in our new configuration. As an example, we choose d₁ = 25 μm. Note that the TE₁ mode peaks at the centre of the waveguide, inside of the high-index (n₂) material, while the TE₃ mode also has peaks inside of the low-index (n₁) material. It is also seen that changing the index from 1.53 to 1.7 over part of the slab does not completely change the mode shape. Therefore, one expects that if the index of refraction in the region that covers the central lobe of the TE₃ mode is increased to 1.7, it will increase the effective index of the TE₃ mode, and it will also increase the index of the TE₁ mode, because approximately 60% of the energy in the TE₁ mode is the center 1/3 of the slab, while only 33% of the energy is in the TE₃ mode. For a waveguide without the graphene, we have expanded the transcendental equation for the mode propagation constant in terms of the index difference and obtain the following approximate expression for the effective index of refraction for the TE_n mode:

n_{e ff}^{(n)} = n_{e ff}^{o} + \frac{n_{1}}{n_{e ff}^{o}} () \frac{\sin (2 π n \frac{d_{1}}{b})}{n π} + 1 - \frac{2 d_{1}}{b})) (n_{2} - n_{1}),

(24)

where $n_{e ff}^{o}$ is the effective index of refraction for the TE_n mode for the original structure. Note that when there is no graphene, $n_{e ff}^{o}$ is the same for the TE₁ mode at ω and the TE₃ mode at 3ω. As can be seen, when d₁ = 0 and d₁ = b/2, $n_{e ff}^{(n)}$ is the same for both modes. However, if n₂ > n₁, then for all other values of d₁, $n_{e ff}^{(3)} < n_{e ff}^{(1)}$ ⁠, which is the desired effect.

FIG. 6.

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Absolute value of the electric field for the normalized TE₁ mode at ω and the TE₃ mode at 3ω at z = 0 for a 2 THz field in the new configuration (n₁ = 1.53 and n₂ = 1.70) and original configuration (n₁ = 1.53 and n₂ = 1.53) for d₁ = 25 μm and E_F = 50 meV.

We now use the exact model to examine how the effective refractive index changes with the thickness of the region with lower refractive index (d₁). In Fig. 7, we plot the linear effective refractive index difference, Δn_eff, for a waveguide with plate separation of b = 70 μm for different Fermi energies as a function of d₁. In this calculation, we set $n_{2}^{(1)} (ω)$ and $n_{2}^{(3)} (3 ω)$ to zero. It is seen that perfect phase matching (Δn_eff ≡ 0) occurs at two points for each value of E_F as we increase d₁. More importantly, we see that we can reduce Δn_eff by up to 0.1, which is more than enough to compensate for the effective index difference shown in Fig. 3 over the full range of Fermi energies, up to incident fields of at least 15 kV/cm. For comparison, we also plot the result using the approximate expression of Eq. (24) when there is no graphene; as can be seen, this equation gives a result very similar to the graphene with low doping but with a small shift in the location of the minimum arising from the lowest-order expansion approximation. This new configuration can be used to not only help to decrease the phase mismatching induced by the SFM and XFM but overcome the linear phase mismatch introduced by the graphene. Note also that the new configuration leads to an increase in the amplitude of the field at the graphene in the TE₁ mode. This will yield a slight increase in the generated field for a given input power.

FIG. 7.

$FIG. 7. Effective refractive index difference (Δneff) between the TE3 mode at 3ω and the TE1 mode at ω as a function of d1 for Fermi energies of EF = 20 meV, EF = 50 meV, and EF = 100 meV using the exact expression and also EF = 0 using the approximate expression of Eq. (24). This is calculated without XFM and SFM, i.e., we set n2(1)(ω)=0 and n2(3)(3ω)=0 in Eqs. (21) and (22).$

View large Download slide

Effective refractive index difference (Δn_eff) between the TE₃ mode at 3ω and the TE₁ mode at ω as a function of d₁ for Fermi energies of E_F = 20 meV, E_F = 50 meV, and E_F = 100 meV using the exact expression and also E_F = 0 using the approximate expression of Eq. (24). This is calculated without XFM and SFM, i.e., we set $n_{2}^{(1)} (ω) = 0$ and $n_{2}^{(3)} (3 ω) = 0$ in Eqs. (21) and (22).

B. Power efficiency

In Fig. 8, we plot the power efficiency of the waveguide in our new configuration as a function of d₁ for an incident field of 5 kV/cm at Fermi energies of E_F = 20 meV, E_F = 50 meV, and E_F = 100 meV. Decreasing the Fermi energy leads to higher power efficiency due to the increase in the nonlinear conductivity²³ and reduced loss due to a decrease in the linear conductivity.^9,22 Note that when d₁ = 35 μm, our new configuration PPW is identical to our original PPW, and so we obtain the same efficiency as is given in Fig. 4(a). At this low input power, we know from Fig. 3 that the effects of XFM and SFM are almost negligible except for E_F = 20 meV. Therefore, we expect to obtain the peaks in the efficiency close to the values of d₁ where Δn_eff is zero in Fig. 7. This is indeed what we find, but with small shifts that arise from the need to also compensate for XFM and SFM. As expected from Fig. 3, this shift is largest for the Fermi energy of 20 meV. We also note that for this small input field, the increase in the efficiency over the initial-configuration PPW is rather modest (<30%).

FIG. 8.

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Power efficiency of the new configuration as a function of d₁ for the full calculation for three different Fermi energies and an input field of E_input = 5 kV/cm.

In Fig. 9, we plot the power efficiency of the waveguide in the new configuration as a function of d₁ for a Fermi energy of 20 meV for input fields of 5, 10, and 15 kV/cm. As shown inFig. 8, the efficiency peaks at two different values for each input field. However, for the higher input fields, we see that these peaks are much larger and have shifted to values of d₁ that are closer to the minimum in Δn_eff shown in Fig. 7. Both of these effects are the result of the need to compensate for the much larger phase mismatch that arises from XFM and SFM at high input fields.

FIG. 9.

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Power efficiency in the new configuration as a function of d₁ for the full calculation for a Fermi energy of E_F = 20 meV for three different input fields.

In Fig. 10, we compare the power efficiency of the waveguide in the original configuration with and without SFM and XFM to the efficiency of the waveguide in the new configuration with the full calculation. In the new configuration, the optimized power efficiency is improved such that it equals or improves upon the results we obtained for the original configuration when SFM and XFM are neglected. This is because our new configuration overcomes not only the phase mismatch induced by SFM and XFM but also that induced by the linear response of the graphene. For example, for E_F = 20 meV and E_input = 10 kV/cm, the power efficiency increases from 28% to 48%, and for E_input = 15 kV/cm, it increases from 35% to 75%.

FIG. 10.

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Power efficiency as a function of Fermi energy for three different input fields of (a) 5, (b) 10, and (c) 15 kV/cm. The dashed curves are for the original PPW with (blue) and without (green) XFM and XFM. The solid curve is the result found including XFM and SFM for the optimized new configuration PPW.

Due to the experimental difficulties in achieving a uniform doping over graphene sheets that are millimetres in length, achieving low Fermi energies is very challenging. Therefore, we now examine what efficiencies can be achieved for higher Fermi energies if we move to higher input fields. In Fig. 11(a), we plot the optimized power efficiency of the waveguide in the new configuration as a function of the input field for different Fermi energies. Note that the optimized length and d₁ are different for each input field and Fermi energy [see Figs. 11(b) and 11(c)]. We note that for all three Fermi energies, the efficiency initially rises with input power, but it then reaches a peak and settles to a value of around 30% at high input fields. To understand this, consider Fig. 3, where we see that for E_F = 20 meV, the index mismatch due to XFM and SFM is already 0.06 at an input field of 15 kV/cm. It is easy to see therefore that for the fields considered inFig. 10, we will quickly reach index differences greater than 0.1 which is the maximum that can be compensated for using our new configuration PPW. Therefore, the input field at which the efficiency peaks for a given Fermi energy is the field at which the nonlinear effective index difference reaches about 0.1. Due to the strong dependence of XFM and SFM on the Fermi energy, this field amplitude is different for the different Fermi energies. The highest efficiencies are obtained for the lowest Fermi energy of 20 meV, largely because the loss is lower in this system and because the higher nonlinearity means that the structure length is less. Note that for all three Fermi energies, the peak efficiencies occur for devices with a length of only a few hundred microns. The reason that the high-field efficiency is essentially independent of Fermi energy is because in all cases, the pump is depleted over a distance that is less than the linear loss distance (1.2 mm, 0.48 mm, and 0.24 mm for Fermi energies of 20 meV, 50 meV, and 100 meV, respectively) and so the different losses at the different Fermi energies do not play a significant role. Therefore, we find that very good efficiencies can be obtained for higher Fermi energies if one can attain the higher input fields. For example, for a Fermi energy of 100 meV, we are able to obtain a power efficiency of 30% at an input field of 50 kV/cm. Because at high-fields, our structure is not able to compensate for SFM and XFM, we find (not shown) that efficiencies of up to 40% can be obtained at high fields and large Fermi energies even in our original configuration. This is a very promising configuration that we believe should be achievable in the lab using current graphene samples and THz sources.^13,24

FIG. 11.

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(a) Power efficiency of the waveguide in the new configuration as a function of the input field for Fermi energies of E_F = 20, 50, and 100 meV. [(b) and (c)] Optimized length and d₁ as a function of input field.

V. SUMMARY

We have developed a coupled-mode theory for the propagating lossy modes of the pump and third harmonic fields in a PPW to calculate third harmonic generation, including pump depletion, SFM, and XFM. We find that SFM and XFM degrades the phase matching between the TE₁ mode at ω and the TE₃ mode at 3ω and thereby decreases the generated third-harmonic electric field. We have shown that one can overcome the phase mismatch due to SFM and XFM by designing a new configuration PPW. We found that by optimizing the dielectric layer thickness, the power efficiency can be increased by more than a factor of two relative to the original configuration. We have also shown that even for graphene with Fermi energy of 100 meV, where the nonlinearity is relatively modest, efficiencies of up to 30% can be achieved for input field amplitudes of 50 kV/cm. We therefore believe that our PPW system is an excellent platform to produce and examine harmonic generation in monolayer graphene.

ACKNOWLEDGMENTS

We thank the Natural Sciences and Engineering Research Council of Canada and Queen’s University for financial support. The authors would like to thank Lukas Helt for useful discussions.

APPENDIX: DYNAMIC EQUATIONS OF THE LINEAR AND THIRD-HARMONIC ELECTRIC FIELD

In this appendix, we give the details of the derivation of our coupled-mode equations (18) and (19).

The LHS of Eq. (17) can be written as

\nabla^{2} E (y, z; 3 ω) = \frac{d^{2} E (y, z; 3 ω)}{d y^{2}} + \frac{d^{2} E (y, z; 3 ω)}{d z^{2}} .

(A1)

If we define $S_{n}^{(1)} () y)) = \sin ({\tilde{k}}_{n} (ω) y)$ ⁠,

\frac{d^{2} E (y, z; 3 ω)}{d y^{2}} = \sum_{n} A_{n} (z, 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{'' (1)} (y),

(A2)

where

S_{n}^{'' (1)} () y)) = - {\tilde{k}}_{n}^{2} (ω) S_{n}^{(1)} () y)) .

(A3)

However, Eq. (A3) is not valid at y = b/2. According to the results obtained for the first derivative of the electric field at the graphene, as shown in Fig. 12, we must add a term to our calculations for the electric field at the graphene. This term is given by Cδ(y − b/2) such that

S_{n}^{'' (1)} () y)) = - {\tilde{k}}_{n}^{2} (3 ω) S_{n}^{(1)} () y)) + C δ (y - b / 2) .

(A4)

To determine C, we take integrate $S_{n}^{'' (1)} () y))$ from y = b/2 − ϵ to y = b/2 + ϵ, for ϵ ≪ b. Then, we obtain

C = \int_{b / 2 - ϵ}^{b / 2 + ϵ} S_{n}^{'' (1)} () y)) d y = S_{n}^{' (1)} () y = {\frac{b}{2}}^{+})) - S_{n}^{' (1)} () y = {\frac{b}{2}}^{-})) .

(A5)

The electric field below and above the graphene is defined as

\begin{matrix} \begin{matrix} E_{b e l o w}^{(n)} (y, z; 3 ω) = & E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} (3 ω) y) \hat{x} 0 < y < b / 2, \\ E_{a b o v e}^{(n)} (y, z; 3 ω) = & E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} (3 ω) (y - b)) \hat{x} b / 2 < y < b, \end{matrix} \end{matrix}

(A6)

and using Maxwell’s equations, we have

\begin{matrix} \nabla \times E_{b e l o w}^{(n)} (y, z; 3 ω) = & - (- i 3 ω) μ_{0} H_{b e l o w}^{(n)} (y, z; 3 ω), \\ \nabla \times E_{a b o v e}^{(n)} (y, z; 3 ω) = & - (- i 3 ω) μ_{0} H_{a b o v e}^{(n)} (y, z; 3 ω) . \end{matrix}

(A7)

Thus, Eq. (A7) can be written as

\begin{matrix} H_{b e l o w}^{(n)} (y, z; 3 ω) & = & \frac{{\tilde{β}}_{n} (3 ω)}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} y) \hat{y} \\ + \frac{i}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \frac{d \sin ({\tilde{k}}_{n} y)}{d y} \hat{z}, \\ H_{a b o v e}^{(n)} (y, z; 3 ω) & = & - \frac{{\tilde{β}}_{n} (3 ω)}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} (y - b)) \hat{y} \\ - \frac{i}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \frac{d \sin ({\tilde{k}}_{n} (y - b))}{d y} \hat{z} . \end{matrix}

(A8)

The current density at the graphene is given by

J_{L} (z; 3 ω) = \sum_{n} J_{L n} (z; 3 ω),

(A9)

where the current density at the graphene for the nth mode is related to the field by

\begin{align} J_{L n} (z; 3 ω) = & - \frac{i}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \frac{d \sin ({\tilde{k}}_{n} (y - b))}{d y} \hat{x} |_{y = {\frac{b}{2}}^{+}} \\ - \frac{i}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \frac{d \sin ({\tilde{k}}_{n} y)}{d y} \hat{x} |_{y = {\frac{b}{2}}^{-}}, \end{align}

(A10)

where

\frac{d \sin ({\tilde{k}}_{n} (y - b))}{d y} |_{y = {\frac{b}{2}}^{+}} = - \frac{d \sin ({\tilde{k}}_{n} y)}{d y} |_{y = {\frac{b}{2}}^{-}} .

(A11)

Thus, Eq. (A10) can be written as

\begin{matrix} J_{L n} (z; 3 ω) = \frac{- i}{3 ω μ_{0}} E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \{) S_{n}^{' (1)} () y = {\frac{b}{2}}^{-})) - S_{n}^{' (1)} () y = {\frac{b}{2}}^{+})) \}) \hat{x} . \end{matrix}

(A12)

The current density at the graphene is related to the linear conductivity by

\begin{matrix} J_{L} (z; 3 ω) = & \sum_{n} J_{L n} (z; 3 ω) \\ = & σ^{(1)} (3 ω) E_{n} (z; 3 ω), \end{matrix}

(A13)

where

E_{n} (z; 3 ω) = E_{n} e^{i {\tilde{β}}_{n} (3 ω) z} \sin ({\tilde{k}}_{n} \frac{b}{2}) \hat{x} .

(A14)

Equality of Eqs. (A12) and (A13) gives

S_{n}^{' (1)} () y = {\frac{b}{2}}^{-})) - S_{n}^{' (1)} () y = {\frac{b}{2}}^{+})) = 3 i ω μ_{0} σ^{(1)} (3 ω) S_{n}^{(1)} () \frac{b}{2})) .

(A15)

Substituting Eq. (A15) into Eq. (A5), we obtain

C = 3 i ω μ_{0} σ^{(1)} (ω) S_{n}^{(1)} () \frac{b}{2})) .

(A16)

Thus, Eq. (A4) becomes

S_{n}^{'' (1)} () y)) = - {\tilde{k}}_{n}^{2} S_{n}^{(1)} () y)) - 3 i ω μ_{0} σ^{(1)} (3 ω) S_{n}^{(1)} () \frac{b}{2})),

(A17)

and Eq. (A2) can be written as

\begin{align} \frac{d^{2} E (z, y; 3 ω)}{d y^{2}} = & - \sum_{n} \{) {\tilde{k}}_{n}^{2} A_{n} (z; 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(1)} () y)) \\ + 3 i ω μ_{0} σ^{(1)} (3 ω) A_{n} (z; 3 ω) S_{n}^{(1)} () y = \frac{b}{2})) \}) . \end{align}

(A18)

Using the slowly varying envelope approximation, where we neglect the second derivative of the envelope, we have

\begin{matrix} \frac{d^{2} E (z, y; 3 ω)}{d z^{2}} & = & \sum_{n} \{) 2 i {\tilde{β}}_{n} (3 ω) \frac{d A_{n} (z; 3 ω)}{d z} \\ - {\tilde{β}}_{n}^{2} (3 ω) A_{n} (z; 3 ω) \}) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(1)} (y) . \end{matrix}

(A19)

If we use Eqs. (A18) and (A19) in Eq. (17) of the main text and retain only those terms that are maximally phase matched, then we obtain

\begin{matrix} - \sum_{n} 2 i {\tilde{β}}_{n} (3 ω) \frac{d A_{n} (z; 3 ω)}{d z} e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(3)} () y)) + \sum_{n} \{) {\tilde{β}}_{n}^{2} (3 ω) + {\tilde{k}}_{n}^{2} (3 ω) - 9 μ_{0} ϵ ω^{2} \}) A_{n} (z; 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(3)} () y)) \\ + \sum_{n} 3 i ω μ_{0} σ^{(1)} (3 ω) A_{n} (z; 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(3)} () y = b / 2)) δ (y - b / 2) \\ = \sum_{n} 3 i ω μ_{0} σ^{(1)} (3 ω) A_{n} (z; 3 ω) e^{i {\tilde{β}}_{n} (3 ω) z} S_{n}^{(3)} () y)) δ (y - b / 2) + 3 i ω μ_{0} σ^{(3)} (3 ω; ω, ω, ω) \sum_{n^{'}} \{) A_{n^{'}} (z; ω) {\})}^{3} e^{3 i {\tilde{β}}_{n^{'}} (ω) z} \{) S_{n^{'}}^{(1)} () y)) {\})}^{3} \\ \times δ (y - b / 2) + 3 i ω μ_{0} σ^{(3)} (3 ω; 3 ω, - 3 ω, 3 ω) \sum_{n} A_{n} (z; 3 ω) | A_{n} (z; 3 ω) |^{2} e^{i (2 {\tilde{β}}_{n} (3 ω) - {\tilde{β}}_{n}^{*} (3 ω)) z} S_{n}^{* (3)} () y)) \{) S_{n}^{(3)} () y)) {\})}^{2} δ (y - b / 2) \\ + 6 i ω μ_{0} σ^{(3)} (3 ω; 3 ω, - ω, ω) \sum_{n} \sum_{n^{'}} A_{n} (z; 3 ω) | A_{n^{'}} (z; ω) |^{2} e^{i ({\tilde{β}}_{n} (3 ω) + {\tilde{β}}_{n^{'}} (ω) - {\tilde{β}}_{n}^{*} (ω)) z} S_{n}^{(3)} () y)) | S_{n^{'}}^{(1)} () y)) |^{2} δ (y - b / 2) . \end{matrix}

(A20)

For a lossy waveguide, the propagation constant is defined as ${\tilde{β}}_{n} (3 ω) = \sqrt{9 μ_{0} ϵ ω^{2} - {\tilde{k}}_{n}^{2} (3 ω)}$ ⁠. Using this removes the second term in Eq. (A20). We now multiply Eq. (A20) by $S_{m}^{* (1)} () y))$ and integrate over y, using

\begin{matrix} \int_{0}^{b} {S_{m}^{*}}^{(1)} () y)) S_{n}^{(1)} () y)) d y ≃ b / 2 δ_{n, m} \\ \int_{0}^{b} {S_{m}^{*}}^{(1)} () y)) S_{n}^{(1)} () y)) δ (y - b / 2) d y = {S_{m}^{*}}^{(1)} () \frac{b}{2})) S_{n}^{(1)} () \frac{b}{2})) \\ \int_{0}^{b} {S_{m}^{*}}^{(1)} () y)) S_{n^{'}}^{(3)} () y)) \{) S_{n}^{* (1)} () y)) {\})}^{2} δ (y - b / 2) d y \\ = {S_{m}^{*}}^{(1)} () \frac{b}{2})) S_{n^{'}}^{(3)} () \frac{b}{2})) \{) S_{n}^{* (1)} () \frac{b}{2})) {\})}^{2} . \end{matrix}

(A21)

From this, we obtain Eq. (18) for the differential equation of the amplitude of the electric field at 3ω for the mth mode. A similar calculation yields Eq. (19) for the differential equation for the electric field at ω. Note that we have confirmed numerically that the slowly varying envelope approximation, where we neglect the second derivatives of the envelopes, is an excellent approximation for all fields and Fermi energies.

FIG. 12.

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(a) Electric field and (b) first derivative of the electric field for the TE₁ mode for a 2 THz field in the waveguide in the presence of graphene for a plate separation of b = 70 μm. The discontinuity in the mode profile is due to the surface current at the graphene.

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Author(s)

Optimized nonlinear terahertz response of graphene in a parallel-plate waveguide

I. INTRODUCTION

II. THEORY

A. Linear modes

B. Coupled mode equations

III. RESULTS

IV. NEW CONFIGURATION: MITIGATING SELF- AND CROSS-PHASE MODULATION

A. Phase matching

B. Power efficiency

V. SUMMARY

ACKNOWLEDGMENTS

APPENDIX: DYNAMIC EQUATIONS OF THE LINEAR AND THIRD-HARMONIC ELECTRIC FIELD

REFERENCES

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Optimized nonlinear terahertz response of graphene in a parallel-plate waveguide

I. INTRODUCTION

II. THEORY

A. Linear modes

B. Coupled mode equations

III. RESULTS

IV. NEW CONFIGURATION: MITIGATING SELF- AND CROSS-PHASE MODULATION

A. Phase matching

B. Power efficiency

V. SUMMARY

ACKNOWLEDGMENTS

APPENDIX: DYNAMIC EQUATIONS OF THE LINEAR AND THIRD-HARMONIC ELECTRIC FIELD

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

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