Transition radiation from graphene plasmons by a bunch beam in the terahertz regime

Kai-Chun Zhang; Xiao-Xing Chen; Chang-Jian Sheng; Kelvin J. A. Ooi; Lay Kee Ang; Xue-Song Yuan

doi:10.1364/OE.25.020477

1. Introduction

Plasmons are the collective excitations of conduction electrons at the metal’s surface [1]. They can be excited either by an incident electromagnetic wave or by an electron beam on the metal-dielectric interface, to form coupled oscillations called surface plasmon polaritons (SPPs) [2,3]. The confinement of SPPs on the metal-dielectric interface not only allows propagation of light below the diffraction limit, it also enhances radiative emission from the surface as well. One of the plasmonic materials available today is graphene. Different from metals, the electrons in graphene are uniquely two-dimensional massless fermions that exhibit exceptional electronic properties [4], offering unduly performance enhancement in photonics, optoelectronics and plamonics [5–9]. Also, unlike in noble metals, plasmonic excitations in graphene can be tuned by shifting graphene’s Fermi energy through chemical doping or electrostatic bias. Within the carrier density of lowly-doped graphene, the graphene plasmon frequency lies right at the terahertz (THz) frequency range. As such, electrons moving uniformly and parallel to the graphene sheet on a substrate are predicted to generate SPP-enhanced THz radiation [10–16]. Further, inelastic electron tunneling has been proposed as a low-energy pathway for the highly-efficient excitation of surface plasmons at the THz frequencies [17]. On the other hand, when electrons are in normal or oblique incidence to the graphene surface, both SPP and transition radiation (TR) will be generated [18]. The TR is determined by the polarization current of the graphene’s surface, which is influenced by the plasmons, and therefore are dynamically tunable through electrostatic gating. The ability to effectively generate tunable coherent THz waves from graphene has recently became one of the most attractive subjects, as they have wide applications in bio-imaging, biomolecule identification, security inspection and ultrafast wireless communications.

In this work, we propose a tunable THz coherent radiation source based on graphene SPP-induce TR. From classical electromagnetic theory, we take into account the polarization current of the graphene’s surface to study the TR from graphene. The properties of THz TR are analyzed for the case when a travelling electron-bunch is at normal incidence to the graphene boundary. In Section 2, we theoretically study the field distribution and spectral-angular distribution of TR excited by a bunch. Then, in Section 3, we present the calculation results and discussion.

2. Transition radiation from graphene by a bunch

With the remarkable development of ultrafast technology, emission from an electron beam can now be precisely controlled. Recently, a TR-based THz source with energies in the order of μJ/pulse has been reported [19]. This coherent TR is produced by laser-driven electron bunches across a vacuum-solid boundary, with a reasonable number of electrons (N_e ~1.32 × 10¹⁰) in the bunch.

When a bunched beam transmits through a graphene layer on a dielectric substrate, it excites the in-plane graphene SPPs. Then, the polarization current density induced by SPPs will result in out-of-plane TR. The induced two-dimensional polarization current density J_// is given by σE_// according to Ohm’s law where σ is the conductivity of graphene and E_// is the in-plane electric field.

A. Radiation field

Let us consider a graphene layer as an infinitely thin conducting sheet placed at the interface (z = 0) between vacuum (z>0) and a dielectric substrate (z<0) with the relative permittivity ε₁ and ε₂, respectively, as shown in Fig. 1. When an electron bunch normally passes through the graphene monolayer, it induces a polarization current density, which serves as source of TR.

Fig. 1 Schematic illustration of the system.

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The Maxwell’s wave equation can be written for Fourier images of electric field and polarization current density as follows

\nabla^{2} E (r, ω) + \frac{ω^{2}}{c^{2}} ε_{r} E (r, ω) = \frac{\nabla (ρ (r, ω))}{ε_{0} ε_{r}} + i ω μ J (r, ω)

For an electron bunch, the charge density and current density in time domain can be found from

ρ (r, t) = e \sum_{j} v_{j} δ (r_{j} - v_{j} t - r_{0}_{j}), J (r, t) = e \sum_{j} v_{j} δ (r_{j} - v_{j} t - r_{0}_{j})

where e is charge of electron, the subscript j denotes the j^th electron, and r_0j and v_j are the position vector and the velocity of the j^th electron, respectively.

Each electron in the bunch can be described as an object in a six-dimensional position-momentum space, which consists of a three-dimensional position distribution and a three-dimensional momentum distribution. For simplicity, we will assume that there is no correlation between the position and the momentum of electrons and all electrons move with the same velocity. Following this assumption, the bunch distribution can be reduced to a three-dimensional position distribution. Substituting the Fourier transform of the charge density and the beam current density in Eq. (2) into the Fourier transform of the wave equation in Eq. (1), the Fourier image of electric field accompanied by a traveling bunch with velocity v is now written as

E_{0} (Q, ω) = \frac{i e}{ε_{0} ε_{r}} \frac{ε_{r} ω v - c^{2} Q}{c^{2} Q^{2} - ω^{2} ε_{r}} \sum_{j} e^{- i Q_{/ /} • r_{/ / j} - i Q_{z} z_{j}}

where

Q = {Q_{/ /} e_{/ /} Q_{z} e_{z}}

is the wave vector,

Q_{z} = \frac{ω}{v_{z}}

, and

v = {0 0 - v_{z}}

, as shown in Fig. 1. The subscripts // and z represent the direction parallel with the plane of graphene layer and the direction of the bunch trajectory, respectively.

Subsequently, the magnetic field of a travelling electron bunch is derived as

H_{0 φ} (Q_{/ /}, z, ω) = i e c^{2} \frac{- Q_{/ /}}{c^{2} Q^{2} - ω^{2} ε_{r}} e^{i Q_{z} z} \sum_{j} e^{- i Q_{/ /} • r_{/ / j} - i Q_{z} z_{j}} e_{φ}

where the subscript φ denotes the angle from graphene plane.

The TR fields can be calculated by solving the wave equations in both the vacuum and substrate and by applying the continuity condition of the tangential electric and magnetic fields across the boundary of the graphene layer. The complete solutions of the wave equations contain both the charged particle fields and the homogeneous TR fields. From the continuity equations across the interface (z = 0), according to the fact that the homogeneous fields are convergent, $k • E = 0$ , the homogeneous radiation fields induced by graphene in vacuum (z>0, zone I) and substrate (z<0, zone II) are written as:

{\begin{cases} E_{r / /}^{I} (k_{/ /}, z, ω) = A e^{i k_{z}^{I} z} e_{/ /} \\ H_{r φ}^{I} = \frac{ε_{0} ε_{1} ω}{k_{z}^{I}} A e^{i k_{z}^{I} z} e_{φ} \end{cases} and {\begin{cases} E_{r / /}^{I I} (k_{/ /}, z, ω) = B e^{- i k_{z}^{I I} z} e_{/ /} \\ H_{r φ}^{I I} = - \frac{ε_{0} ε_{2} ω}{k_{z}^{I I}} B e^{- i k_{z}^{I I} z} e_{φ} \end{cases}

where

k_{z}^{I (I I)} = \sqrt{\frac{ω^{2}}{c^{2}} ε_{1 (2)} - k_{/ /}^{2}}

is the propagation wave vector. Factors A and B are determined by the boundary conditions on the vacuum-substrate interface:

n \times (E_{2} - E_{1}) = 0

and

n \times (H_{2} - H_{1}) = J_{/ /}

, and

E_{1 (2)}

and

H_{1 (2)}

are the total electric and magnetic field in the zone I(II), respectively.

The spatial distribution of the electron beam can be considered as a Gaussian distribution through $f = {[{(2 π)}^{3 / 2} σ_{r}^{2} σ_{z}]}^{- 1} \exp (- r_{/ /}^{2} / 2 σ_{r}^{2}) \exp (- z^{2} / 2 σ_{z}^{2})$ , where σ_z and σ_r are the root-mean-square longitudinal length of beam duration and transverse width, respectively. By solving Eq. (2-5) with the boundary conditions, we get the factors A and B of the radiation field as

{\begin{cases} A = A_{0} \sum_{j}^{N} e^{- i Q_{/ /} • r_{/ /_{j}} - i Q_{z} z_{j}} = A_{0} N F (k_{/ /}, Q_{z}) \\ B = B_{0} \sum_{j}^{N} e^{- i Q_{/ /} • r_{/ /}_{j} - i Q_{z} z_{j}} = B_{0} N F (k_{/ /}, Q_{z}) \end{cases}

where

\begin{array}{l} {\begin{cases} A_{0} = \frac{\frac{i e c^{2} k_{z}^{I} k_{/ /}}{v_{z} ε_{1}} [\frac{ω ε_{2} + v_{z} k_{z}^{I I} ε_{1} + σ_{g} k_{z}^{I I} / ε_{0}}{c^{2} Q^{2} - ω^{2} ε_{1}} - \frac{ω ε_{1} + v_{z} k_{z}^{I I} ε_{1}}{c^{2} Q^{2} - ω^{2} ε_{2}}]}{k_{z}^{I} ε_{2} ω ε_{0} + k_{z}^{I I} ε_{1} ω ε_{0} + σ_{g} k_{z}^{I} k_{z}^{I I}} \\ B_{0} = \frac{- \frac{i e c^{2} k_{z}^{I I} k_{/ /}}{v_{z} ε_{2}} [\frac{ω ε_{2} - v_{z} k_{z}^{I} ε_{2}}{c^{2} Q^{2} - ω^{2} ε_{1}} - \frac{ω ε_{1} - v_{z} k_{z}^{I} ε_{2} + σ_{g} k_{z}^{I} / ε_{0}}{c^{2} Q^{2} - ω^{2} ε_{2}}]}{k_{z}^{I} ε_{2} ω ε_{0} + k_{z}^{I I} ε_{1} ω ε_{0} + σ_{g} k_{z}^{I} k_{z}^{I I}} \end{cases} and \\ F (k_{/ /}, Q_{z}) = \exp (- \frac{k_{/ /}^{2} σ_{r}^{2} + Q_{z}^{2} σ_{z}^{2}}{2}) . \end{array}

σ_g is the intraband conductivity of graphene monolayer, A₀ and B₀ are the amplitudes of the radiation field excited by a single electron, N is the number of electrons, and F = F_//F_z is the spatial factor which is closely related to the Fourier transform of the electron beam’s spatial distribution. For a Gaussian spatial beam distribution, the spatial factor is expressed by

F_{/ /} = \exp (- \frac{1}{2} k_{/ /}^{2} σ_{r}^{2}) and F_{z} = \exp (- \frac{1}{2} Q_{z}^{2} σ_{z}^{2}) .

When the denominators of A₀ and B₀ are equal to zero, the corresponding k_// will represent the wave vector of the graphene SPP. One can see that when the evanescent waves from an incident electron bunch excites the SPPs on graphene layer, it will spontaneously induce a polarization current density according to the Drude conductivity and the graphene plasmon frequency, which serves as the source of the TR at the same frequency. After straightforward mathematical manipulations, the dispersion of the graphene SPPs on the interface can be written as:

\frac{ε_{1}}{\sqrt{k_{/ /}^{2} - \frac{ε_{1} ω^{2}}{c^{2}}}} + \frac{ε_{2}}{\sqrt{k_{/ /}^{2} - \frac{ε_{2} ω^{2}}{c^{2}}}} = - i \frac{σ_{g}}{ω ε_{0}}

Within linear response theory, graphene’s dynamic conductivity is the sum of interband electronic transitions and intraband Drude-like processes. However, in the THz range, the intraband contribution plays a dominant role in the optical conductivity. Thus a Drude-like intraband conductivity of graphene layer can be obtained [20]

σ_{g} = \frac{e^{2} E_{F}}{π ℏ^{2}} \frac{i}{ω + i τ^{- 1}} = \frac{e^{2} E_{F}}{π ℏ^{2}} \frac{τ^{- 1} + i ω}{ω^{2} + τ^{- 2}}

where τ is the carrier relaxation time, and

E_{F} = ℏ v_{F} \sqrt{π n_{s}}

is the Fermi energy, which is determined by carrier concentration n_s and Fermi velocity v_F.

Finally, the electric field of TR in the graphene plane can be expressed as

\begin{array}{l} E_{r / /} (r, ω) = \frac{1}{2 π} \int E_{r 0 / /} (k_{/ /}, z, ω) \sum_{j}^{N} e^{- i Q_{/ /} • r_{/ / j} - i Q_{z} z_{j}} e^{i k_{/ /} • r_{/ /}} d k_{/ /} \\ = \int_{0}^{\infty} k_{/ /} A_{0} e^{i k_{z}^{r} z} J_{0} (k_{/ /} r_{/ /}) N F (k_{/ /}, Q_{z}) d k_{/ /} \end{array}

where

E_{r 0}

is the radiation field caused by a single electron, and J₀ is the Bessel function. When the condition NF>1 is satisfied, the radiation field from the bunched electrons must be larger than that from an individual electron. In this way we can take into account the beam bunching effect on the TR field. Assuming the number of electrons in the bunch to be ~10¹⁰, the bunch parameters are σ_r = 30μm and σ_z = 3μm, which corresponds to a moderate beam energy with a fs pulse.

B. Spectral-angular distribution

The total far-field TR energy from an electron beam through the graphene layer can be obtained from the integral of the Poynting vector

S (r, ω) = \frac{k}{2 ω μ_{0}} [E_{r / /} (r, ω) E_{r / /}^{*} (r, ω) + E_{r z} (r, ω) E_{r z}^{*} (r, ω)]

where

E_{r / /}

and

E_{r z}

are the radiation fields in parallel with the graphene plane and the direction of the bunch trajectory, respectively.

The spectral angular radiation distribution in the frequency domain is given by the expression

W (r, ω) = \int r^{2} n • S (r, ω) d Ω d t

where Ω is the solid angle. One can get the distribution of the TR from graphene over the angles and frequencies by

\frac{d^{2} W (r, ω)}{d Ω d ω} = r^{2} n • S (r, ω) = \frac{k r^{2}}{2 ω μ_{0}} [E_{r / /}^{2} (r, ω) + E_{r z}^{2} (r, ω)]

The summations over electrons in Eq. (9) and Eq. (12) are already taken into account by the three-dimensional electron position distributions. The radiation distribution yields

\frac{d^{2} W (r, ω)}{d Ω d ω} = \frac{r^{2} k}{2 ω μ_{0}} [N E_{0}^{2} + N (N - 1) E_{0}^{2} F^{2}]

where

E_{0}^{2} = E_{0 / /}^{2} (r, ω) + E_{0 z}^{2} (r, ω)

, E_0// and E_0z are the average radiation fields from a single electron parallel to xy-plane and z-plane, respectively. The first term on the right-hand side of Eq. (13) is the contribution from incoherent radiation, which scales as N, and latter term is from coherent radiation, which scales as N². One can see that the coherent radiation is related to the bunch size, while the incoherent radiation is not.

In our calculations, we ignore contributions from Cherenkov radiation in the substrate, Bremsstrahlung as well as characteristic X-rays from graphene or the substrate, and assume graphene is not damaged for electron bunch energies up to 100keV.

3. Results

Given reasonable parameters for graphene’s conductivity, τ = 0.25ps, E_F = 0.1eV, and dielectric constants ε₁ = 1 (z>0), ε₂ = 4 (z<0), we obtained numerical results for the TR field distributions in the r-z plane at 5THz. The results for an individual electron and an electron bunch (N = 10¹⁰, σ_r = 30μm and σ_z = 3μm), both with a beam energy of 3keV, are shown in Fig. 2(a) and 2(b) respectively. From the figures, it is observed that the distribution angles of the radiation fields in vacuum are larger than that in the substrate for both cases. Meanwhile, the amplitude of the radiation fields excited by an electron bunch is up to 10⁷ higher than that by a single electron. Moreover, the radiation angles for bunched electrons are much smaller than that for a single electron. The inset can clearly identify the field contour in the middle part of the figure where the contour is overlapped.

Fig. 2 Electric field of TR (a) by an electron and (b) by an electron bunch at 5THz.

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Next, we perform numerical calculations on the spectral-angular distribution for incoherent and coherent radiations with a beam energy 3keV and r = 500μm, while the other parameters remain the same. The results are shown in Fig. 3. The radiation energy concentrates in a small angular range near θ = 0 (in the backward direction) and θ = π (in forward direction). The intensity of coherent radiation at the lower THz band is far stronger than that of the incoherent one, and the angular distribution of the coherent radiation has two individual frequency peaks near θ = 0 and θ = π respectively. In contrast, incoherent radiation still exists in the higher THz band beyond the graphene plasmon operating frequency.

Fig. 3 Spectral-angular distribution of (a) incoherent and (b) coherent radiation.

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The Poynting fluxes with bunch energies of 1keV, 10keV and 100keV are shown in Fig. 4. The intensity of TR is seen to increase with bunch energy. We also find that the radiation in backward direction increases linearly with the bunch energy, while radiation in forward direction increases nonlinearly. Figure 4(d) shows the TR from 10keV electron bunches with respect to angle for various frequencies, having independent peaks in backward (BW) and forward (FW) directions. At 0.3-0.5THz, the spectral-angular distribution of the radiation intensity mainly lies in the BW direction while at 1.1-1.3THz it lies in the FW direction. In all cases, TR in direction parallel with the graphene layer (θ = π/2) is negligible. The inset in Fig. 4(d) clearly shows the sharp change of the Poynting fluxes at all frequencies near the angle θ = π/2 due to the different dielectric property on two sides of the graphene layer.

Fig. 4 Poynting fluxes with bunch energy (a) 1keV, (b) 10keV, (c) 100keV, respectively, and (d) radiation distribution as a function of angle θ at frequency from 0.3 THz to 1.3 THz.

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We proceed to calculate the TR Poynting flux with varying Fermi energy. In Fig. 5, the three grouped curves, from bottom to top, correspond to the bunch energy of 1keV, 10keV and 100keV respectively. In each group, curves are obtained with varying Fermi energy from 0.1eV to 0.3eV. The peak intensity in the BW direction shown in Fig. 5(a) increases with the Fermi energy and is proportional to the electron bunch energy. The shape of the TR spectrum remains the same throughout the varying Fermi energy and electron bunch energy. In contrast, the shape of the radiation spectrum changes with the electron bunch energy for TR in the FW direction, as shown in Fig. 5(b). The peak intensity decreases with increase of Fermi energy, and the peak frequency shifts upwards with the increase of electron bunch energy. For the electron bunch energy of 100keV, the radiation power in the FW direction is about 400 times higher than that in the BW direction. Overall, these results imply that the radiation intensity and peak frequency can be tuned by external doping (via chemical or electrostatic gating) of graphene. Additionally, as well-known, Bremsstrahlung and characteristic X-rays can be generated from graphene or the substrate by the kilo-voltage energy of electron beam. These radiations mainly distribute in the angle near θ = π/2, while TR mainly in forward and backward, thus they could be detected individually in different azimuth angle.

Fig. 5 TR Poynting flux in (a) backward and (b) forward direction with varying Fermi energy (from 0.1eV to 0.3eV as inset) and bunch energy (1keV, 10keV and 100keV).

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The graphene SPP dispersion is also influenced by the dielectric permittivity of substrate. Here we discuss the effect of dielectric substrate to the TR. For the case when the electron bunch energy is 10keV, the Poynting flux with varying dielectric permittivity in the BW and FW directions is shown in Fig. 6. One can see that, the radiation intensity in the BW direction increases with increase of dielectric constant, while in the FW direction it first increases and then decreases. This is due to the discrepancy between the two boundary conditions for the graphene SPP dispersion for the BW and FW radiation respectively. For the radiation in FW direction, the intensity maximum can be obtained near ε₂ = 7.5 and there are no significant changes for ε₂ from 6 to 10.

Fig. 6 TR Poynting flux in (a) backward and (b) forward direction with varying dielectric constant.

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4. Summary

In this paper, we proposed a tunable coherent THz radiation source based on graphene SPP-induced TR. We present a detailed theoretical analysis of TR in the THz frequency range, which is excited by bunched electrons uniformly passing through a graphene layer with normal incidence. The radiation fields, spectral-angular distribution and Ponyting flux of the TR are derived and numerically analyzed for coherent radiation in the backward and forward directions. The results show that the TR from graphene has unique characteristics arising from the high tunable graphene plasmon dispersion. The frequency of TR can be tuned externally by Fermi energy, dielectric constant of substrate, electron bunch size and energy in the frequency range of 0.4-2THz. The THz TR will find potential uses in THz pulsed imaging (TPI) [21,22], biological imaging [23,24], security and wireless communications applications.

Funding

Fundamental Research Funds for the Central Universities (ZYGX2015J038); National Key Basic Research Program of China (2013CB933603, 2014CB339801); China Scholarship Council (201506075086).

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Transition radiation from graphene plasmons by a bunch beam in the terahertz regime

Abstract

1. Introduction

2. Transition radiation from graphene by a bunch

A. Radiation field

B. Spectral-angular distribution

3. Results

4. Summary

Funding

References and links

Cited By

Figures (6)

Equations (15)

Optics Express