2.1. Overview of simplified PIC-FDTD approach for the analysis of e-beam-induced THz radiation

The PIC-FDTD method has been widely used to study underlying physical mechanism of the Smith-Purcell superradiance [8–10]. To save computational time and memory, we have used simplified version of PIC-FDTD method [17, 18]. In our simplified model, the electron-bunch is treated as one negatively charged particle, the movement of the particle is restricted only in two-dimensional (2D) (x-y) plane, and only transverse electric (TE) mode, with Ex, Ey, and H<sup>z</sup> fields, has been analyzed. Figure 1 shows schematic representation of the analyzed 2D system and definitions of dimensions of the graded grating, where Λ, s, d, and Δd are period, width, depth of the grooves, and depth variation, respectively.

Figure 1. Schematic representation of the analyzed 2D system and definitions of dimensions of the graded grating.

In the FDTD method, the time-dependent EM field propagating in 2D system is simulated using Yee's algorithm [19, 20] to solve the following Maxwell's equations (in the vacuum):

$$\nabla \times \mathbf{E}(\mathbf{x}, y, t) = -\frac{\partial \mathbf{B}(\mathbf{x}, y, t)}{\partial t} \tag{2}$$

CHZLX i þ

CHZLY i þ

1 2 ; j þ 1 2

1 2 ; j þ 1 2

� �

quency dependence of dielectric permittivity of metal can be expressed as follows:

χ ωð Þ¼

εrð Þ¼ ω 1 þ

DðÞ¼ t ε0ε∞Eð Þþ t ε<sup>0</sup>

field E(t) by the convolution:

Figure 2. 2D uniform rectangular Yee's grid for TE mode.

recursive manner.

� �

<sup>¼</sup> <sup>Δ</sup><sup>t</sup> <sup>μ</sup> <sup>i</sup> <sup>þ</sup> <sup>1</sup>

<sup>¼</sup> <sup>Δ</sup><sup>t</sup> <sup>μ</sup> <sup>i</sup> <sup>þ</sup> <sup>1</sup>

Figure 2 shows a typical Yee's 2D uniform rectangular grid for TE mode. E<sup>x</sup> and E<sup>y</sup> components are located at the middle of the edge of each grid, and H<sup>z</sup> component is located at the center of the grids. The time evolution of EM fields is updated in a leapfrog manner. In order to model an open system, we have employed perfectly matched layer (PML)-absorbing conditions [21].

The dielectric properties of metals are strongly dispersive; therefore, we utilized recursive convolution (RC) approach [20] to model metallic grating. By adopting Drude model, fre-

> ω2 p

where ω<sup>p</sup> and Γ are the plasma frequency and collision frequency of metal, respectively. In a linear dispersive medium, the time-domain electric flux density D(t) is related to the electric

> ðt τ¼0

Since the Fourier-transformed electric susceptibility χ(τ) of Drude type of dispersion satisfies the condition for a recursive computation, the convolution in Eq. (15) can be solved in a

ω2 p

<sup>2</sup> ; <sup>j</sup> <sup>þ</sup> <sup>1</sup> 2 � �

<sup>2</sup> ; <sup>j</sup> <sup>þ</sup> <sup>1</sup> 2 � � 1

Electron Beam-Induced Directional Terahertz Radiation from Metamaterials

1

<sup>ω</sup>ð Þ <sup>j</sup><sup>Γ</sup> � <sup>ω</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> χ ωð Þ (13)

<sup>ω</sup>ð Þ <sup>j</sup><sup>Γ</sup> � <sup>ω</sup> (14)

Eð Þ t � τ χ τð Þdτ (15)

<sup>Δ</sup><sup>x</sup> (11)

117

http://dx.doi.org/10.5772/intechopen.80648

<sup>Δ</sup><sup>y</sup> (12)

$$\nabla \times \mathbf{H}(\mathbf{x}, y, t) = \varepsilon\_0 \frac{\partial \mathbf{E}(\mathbf{x}, y, t)}{\partial t} + \mathbf{J}(\mathbf{x}, y, t) \tag{3}$$

where E, H, and B are the electric and magnetic fields and magnetic flux density of EM wave, J is the current density, and ε<sup>0</sup> and μ<sup>0</sup> are the dielectric permittivity and the magnetic permeability in vacuum, respectively. In Yee's algorithm, these differential equations are discretized using centered finite-difference expressions for the space and time derivatives, and we have the following set of equations for TE mode in 2D space:

$$E\_x^{\pi} \left( i + \frac{1}{2}, j \right) = \mathbb{C}\_{\text{EX}} \left( i + \frac{1}{2}, j \right) E\_x^{\pi - 1} \left( i + \frac{1}{2}, j \right) + \mathbb{C}\_{\text{EX} \text{Y}} \left( i + \frac{1}{2}, j \right) \left\{ H\_z^{\pi - \frac{1}{2}} \left( i + \frac{1}{2}, j + \frac{1}{2} \right) - H\_z^{\pi - \frac{1}{2}} \left( i + \frac{1}{2}, j - \frac{1}{2} \right) \right\} \tag{4}$$

$$E\_y^{\mathrm{ul}}\left(i,j+\frac{1}{2}\right) = \mathbb{C}\_{\mathrm{EY}}\left(i,j+\frac{1}{2}\right)E\_y^{\mathrm{u-1}}\left(i,j+\frac{1}{2}\right) + \mathbb{C}\_{\mathrm{EYX}}\left(i,j+\frac{1}{2}\right)\left\{H\_z^{\mathrm{u-\frac{1}{2}}}\left(i+\frac{1}{2},j+\frac{1}{2}\right) - H\_z^{\mathrm{u-\frac{1}{2}}}\left(i-\frac{1}{2},j+\frac{1}{2}\right)\right\},\tag{5}$$

$$H\_z^{n+\frac{1}{2}}\left(i+\frac{1}{2},j+\frac{1}{2}\right) = H\_z^{n-\frac{1}{2}}\left(i+\frac{1}{2},j+\frac{1}{2}\right) - \mathcal{C}\_{\text{HELX}}\left(i+\frac{1}{2},j+\frac{1}{2}\right)\left\{E\_y^n\left(i+1,j+\frac{1}{2}\right) - E\_y^n\left(i,j+\frac{1}{2}\right)\right\}$$

$$+C\_{HZLI}\left(i+\frac{1}{2},j+\frac{1}{2}\right)\left\{E\_x^n\left(i+\frac{1}{2},j+1\right) - E\_x^n\left(i+\frac{1}{2},j\right)\right\}\tag{6}$$

where superscript n is for the time step and (i, j)=(iΔx, jΔy) for the spatial position. The coefficients in Eqs. (4)–(6) are as follows:

$$\mathbf{C}\_{EX}\left(\mathbf{i}+\frac{1}{2},\mathbf{j}\right) = \frac{\mathbf{1}-\frac{\sigma\left(\mathbf{i}+\frac{1}{2}j\right)\Delta t}{2\epsilon\left(\mathbf{i}+\frac{1}{2}j\right)}}{\mathbf{1}+\frac{\sigma\left(\mathbf{i}+\frac{1}{2}j\right)\Delta t}{2\epsilon\left(\mathbf{i}+\frac{1}{2}j\right)}}\tag{7}$$

$$\mathbf{C}\_{\text{EXLY}}\left(\mathbf{i}+\frac{1}{2},\mathbf{j}\right) = \frac{\frac{\Delta t}{\epsilon\left(\mathbf{i}+\frac{1}{2}j\right)}}{1+\frac{\sigma\left(\mathbf{i}+\frac{1}{2}j\right)\Delta t}{2\epsilon\left(\mathbf{i}+\frac{1}{2}j\right)}}\frac{1}{\Delta y} \tag{8}$$

$$\mathbf{C}\_{EY}\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right) = \frac{\mathbf{1} - \frac{\sigma\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)\Delta t}{2\epsilon\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)}}{\mathbf{1} + \frac{\sigma\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)\Delta t}{2\epsilon\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)}}\tag{9}$$

$$\mathbf{C}\_{EYLX}\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right) = \frac{\frac{\Delta t}{\epsilon\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)}}{1+\frac{\sigma\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)\Delta t}{2\epsilon\left(\mathbf{i},\mathbf{j}+\frac{1}{2}\right)}}\frac{1}{\Delta \mathbf{x}}\tag{10}$$

Electron Beam-Induced Directional Terahertz Radiation from Metamaterials http://dx.doi.org/10.5772/intechopen.80648 117

$$\mathbf{C}\_{\rm HZZ} \left( \mathbf{i} + \frac{1}{2}, j + \frac{1}{2} \right) = \frac{\Delta t}{\mu \left( \mathbf{i} + \frac{1}{2}, j + \frac{1}{2} \right)} \frac{1}{\Delta \mathbf{x}} \tag{11}$$

$$\mathbf{C}\_{HZLY}\left(\mathbf{i}+\frac{1}{2},j+\frac{1}{2}\right) = \frac{\Delta t}{\mu\left(\mathbf{i}+\frac{1}{2},j+\frac{1}{2}\right)}\frac{1}{\Delta y} \tag{12}$$

Figure 2 shows a typical Yee's 2D uniform rectangular grid for TE mode. E<sup>x</sup> and E<sup>y</sup> components are located at the middle of the edge of each grid, and H<sup>z</sup> component is located at the center of the grids. The time evolution of EM fields is updated in a leapfrog manner. In order to model an open system, we have employed perfectly matched layer (PML)-absorbing conditions [21].

The dielectric properties of metals are strongly dispersive; therefore, we utilized recursive convolution (RC) approach [20] to model metallic grating. By adopting Drude model, frequency dependence of dielectric permittivity of metal can be expressed as follows:

$$\varepsilon\_r(\omega) = 1 + \frac{\omega\_p^2}{\omega(j\Gamma - \omega)} = 1 + \chi(\omega) \tag{13}$$

$$\chi(\omega) = \frac{\omega\_p^2}{\omega(\text{j}\Gamma - \omega)}\tag{14}$$

where ω<sup>p</sup> and Γ are the plasma frequency and collision frequency of metal, respectively. In a linear dispersive medium, the time-domain electric flux density D(t) is related to the electric field E(t) by the convolution:

$$\mathbf{D}(t) = \varepsilon\_0 \varepsilon\_\approx \mathbf{E}(t) + \varepsilon\_0 \int\_{\tau=0}^t \mathbf{E}(t-\tau)\chi(\tau)d\tau \tag{15}$$

Since the Fourier-transformed electric susceptibility χ(τ) of Drude type of dispersion satisfies the condition for a recursive computation, the convolution in Eq. (15) can be solved in a recursive manner.

Figure 2. 2D uniform rectangular Yee's grid for TE mode.

In the FDTD method, the time-dependent EM field propagating in 2D system is simulated using Yee's algorithm [19, 20] to solve the following Maxwell's equations (in the vacuum):

where E, H, and B are the electric and magnetic fields and magnetic flux density of EM wave, J is the current density, and ε<sup>0</sup> and μ<sup>0</sup> are the dielectric permittivity and the magnetic permeability in vacuum, respectively. In Yee's algorithm, these differential equations are discretized using centered finite-difference expressions for the space and time derivatives, and we have

þ CEXLY i þ

þ CEYLX i; j þ

� CHZLX i þ

where superscript n is for the time step and (i, j)=(iΔx, jΔy) for the spatial position. The

¼

¼

¼

¼

En <sup>x</sup> i þ 1 2 ; j þ 1 

1 2 ; j 

1 2 ; j 

1 2 

1 2  ∂Bð Þ x; y; t

∂Eð Þ x; y; t

1 2 ; j 

1 2 

> 1 2 ; j þ 1 2

<sup>1</sup> � <sup>σ</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup><sup>ε</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup>

<sup>1</sup> <sup>þ</sup> <sup>σ</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup><sup>ε</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup>

Δt <sup>ε</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup> <sup>1</sup> <sup>þ</sup> <sup>σ</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup><sup>ε</sup> <sup>i</sup>þ<sup>1</sup> <sup>2</sup> ð Þ;<sup>j</sup>

 � <sup>σ</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup> <sup>Δ</sup><sup>t</sup> <sup>ε</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>σ</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup> <sup>Δ</sup><sup>t</sup> <sup>ε</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup>

Δt <sup>ε</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>σ</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup> <sup>Δ</sup><sup>t</sup> <sup>2</sup><sup>ε</sup> <sup>i</sup>;jþ<sup>1</sup> ð Þ<sup>2</sup>

H<sup>n</sup>�<sup>1</sup> <sup>2</sup> <sup>z</sup> i þ 1 2 ; j þ 1 2

H<sup>n</sup>�<sup>1</sup> <sup>2</sup> <sup>z</sup> i þ 1 2 ; j þ 1 2

En

� En <sup>x</sup> i þ 1 2 ; j

1

1

<sup>y</sup> i þ 1; j þ

<sup>∂</sup><sup>t</sup> (2)

� <sup>H</sup><sup>n</sup>�<sup>1</sup> <sup>2</sup> <sup>z</sup> i þ 1 2 ; <sup>j</sup> � <sup>1</sup> 2

� <sup>H</sup><sup>n</sup>�<sup>1</sup> <sup>2</sup> <sup>z</sup> <sup>i</sup> � <sup>1</sup> 2 ; j þ 1 2

> � En <sup>y</sup> i; j þ 1 2

<sup>Δ</sup><sup>y</sup> (8)

<sup>Δ</sup><sup>x</sup> (10)

1 2

(4)

(5)

(6)

(7)

(9)

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>J</sup>ð Þ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> (3)

∇ � Eð Þ¼� x; y; t

∇ � Hð Þ¼ x; y; t ε<sup>0</sup>

1 2 

1 2 ; j þ 1 2

CEX i þ

CEXLY i þ

CEY i; j þ

CEYLX i; j þ

the following set of equations for TE mode in 2D space:

En�<sup>1</sup> <sup>x</sup> i þ 1 2 ; j 

E<sup>n</sup>�<sup>1</sup> <sup>y</sup> i; j þ

¼ CEX i þ

116 Metamaterials and Metasurfaces

¼ CEY i; j þ

1 2 ; j 

1 2 

> <sup>¼</sup> <sup>H</sup><sup>n</sup>�<sup>1</sup> <sup>2</sup> <sup>z</sup> i þ 1 2 ; j þ 1 2

þCHZLY i þ

coefficients in Eqs. (4)–(6) are as follows:

En <sup>x</sup> i þ 1 2 ; j 

En <sup>y</sup> i; j þ 1 2 

> H<sup>n</sup>þ<sup>1</sup> <sup>2</sup> <sup>z</sup> i þ 1 2 ; j þ 1 2

In the PIC-FDTD method, time-dependent Maxwell's equations are coupled with the equation of motion of relativistic charged particles driven by the inertia and the Lorentz force and solved in a leapfrog manner similar to the main FDTD algorithm. In our simplified version, we assume the electron-bunch as one negatively charged particle with the following Gaussian spatial charge distribution:

$$m\_{\epsilon}(\mathbf{x}, y, t) = N\_0 \exp\left\{-\frac{(\mathbf{x} - \mathbf{x}\_0(t))^2 + \left(y - y\_0(t)\right)^2}{2\sigma^2}\right\} \tag{16}$$

$$
\sigma = \frac{w}{2\sqrt{2\ln(2)}}\tag{17}
$$

Ex<sup>5</sup> ¼ ð Þ 1 � α Ex<sup>1</sup> þ αEx<sup>2</sup> (20)

Electron Beam-Induced Directional Terahertz Radiation from Metamaterials

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119

Ex<sup>6</sup> ¼ ð Þ 1 � α Ex<sup>3</sup> þ αEx<sup>4</sup> (21)

Ex <sup>¼</sup> <sup>1</sup> � <sup>β</sup> Ex<sup>5</sup> <sup>þ</sup> <sup>β</sup>Ex<sup>6</sup> (22)

<sup>J</sup>ð Þ¼� <sup>x</sup>; <sup>y</sup>; <sup>t</sup> neð Þ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> <sup>e</sup><sup>v</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> (23)

In order to include the movement of the electron-bunch in the FDTD formalism, a current

Figure 4 schematically summarizes our simplified PIC-FDTD simulation scheme. The solution of the time-dependent Maxwell's equations gives spatial counter maps of EM fields and their time evolution. The solution of the equation of motion of relativistic electron-bunch gives its trajectory, and the continuity equation gives the current and charge densities required for

In Figure 1, the analyzed 2D system and definitions of dimensions of the graded grating are schematically shown. In Table 1, parameters used in this study are summarized. The graded grating was assumed to be consisted of Ag, and the Drude model was adopted in order to model the dispersion characters of its dielectric function and solved by using RC scheme as discussed above. The plasma frequency (ωp) and the collision frequency (Γ) of Ag were set to

Maxwell's equations. The flowchart of our PIC-FDTD simulation is shown in Figure 5.

source term is added to Ampere's law:

2.2. Analyzed models and parameters

Figure 4. Schematic representation of simplified PIC-FDTD simulation scheme.

where N<sup>0</sup> is the central electron density of the bunch and w is the half width of the bunch. The coordinate (x0, y0) represents the center of electron-bunch, and its movement (trajectory) is updated by solving following equation of motion:

$$\frac{\partial \mathbf{P}(\mathbf{x}\_0, y\_0, t)}{\partial t} = -n\_t e \left\{ \mathbf{E}(\mathbf{x}\_0, y\_0, t) + \mathbf{v}(\mathbf{x}\_0, y\_0, t) \times \mathbf{B}(\mathbf{x}\_0, y\_0, t) \right\} \tag{18}$$

$$P(\mathbf{x}\_0, y\_0, t) = \frac{m\_\epsilon \sigma(\mathbf{x}\_0, y\_0, t)}{\sqrt{1 - \left|\sigma(\mathbf{x}\_0, y\_0, t)\right|^2/c^2}}\tag{19}$$

where P is the momentum, e is the electron charge, v is the speed of the electron-bunch, and m<sup>e</sup> is the electron mass. The movement of the electron-bunch is assumed to be in a continuous space, and the coordinate (x0, y0) of the center position of it is not necessarily on Yee's discrete FDTD grid points. Therefore, electric field and magnetic flux density on the electron-bunch to solve Eq. (18) should be interpolated from those at the nearest grid points. We used linear interpolation as schematically shown in Figure 3, in which E<sup>x</sup> component is given as follows:

Figure 3. Schematic representation of the linear interpolation for E<sup>x</sup> components at the center position (x0, y0) of the electron-bunch.

Electron Beam-Induced Directional Terahertz Radiation from Metamaterials http://dx.doi.org/10.5772/intechopen.80648 119

$$E\_{\rm x5} = (1 - a)E\_{\rm x1} + aE\_{\rm x2} \tag{20}$$

$$E\_{x6} = (1 - \alpha)E\_{x3} + \alpha E\_{x4} \tag{21}$$

$$E\_x = (1 - \beta)E\_{x5} + \beta E\_{x6} \tag{22}$$

In order to include the movement of the electron-bunch in the FDTD formalism, a current source term is added to Ampere's law:

$$J(\mathbf{x}, y, t) = -n\_e(\mathbf{x}, y, t) \text{ev}\left(\mathbf{x}\_0, y\_0, t\right) \tag{23}$$

Figure 4 schematically summarizes our simplified PIC-FDTD simulation scheme. The solution of the time-dependent Maxwell's equations gives spatial counter maps of EM fields and their time evolution. The solution of the equation of motion of relativistic electron-bunch gives its trajectory, and the continuity equation gives the current and charge densities required for Maxwell's equations. The flowchart of our PIC-FDTD simulation is shown in Figure 5.

#### 2.2. Analyzed models and parameters

In the PIC-FDTD method, time-dependent Maxwell's equations are coupled with the equation of motion of relativistic charged particles driven by the inertia and the Lorentz force and solved in a leapfrog manner similar to the main FDTD algorithm. In our simplified version, we assume the electron-bunch as one negatively charged particle with the following Gaussian

neð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> <sup>N</sup>0exp � ð Þ <sup>x</sup> � <sup>x</sup>0ð Þ<sup>t</sup> <sup>2</sup> <sup>þ</sup> <sup>y</sup> � <sup>y</sup>0ð Þ<sup>t</sup> � �<sup>2</sup>

<sup>σ</sup> <sup>¼</sup> <sup>w</sup> 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>P</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � � <sup>¼</sup> me<sup>v</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � �

where P is the momentum, e is the electron charge, v is the speed of the electron-bunch, and m<sup>e</sup> is the electron mass. The movement of the electron-bunch is assumed to be in a continuous space, and the coordinate (x0, y0) of the center position of it is not necessarily on Yee's discrete FDTD grid points. Therefore, electric field and magnetic flux density on the electron-bunch to solve Eq. (18) should be interpolated from those at the nearest grid points. We used linear interpolation as schematically shown in Figure 3, in which E<sup>x</sup> component is given as follows:

Figure 3. Schematic representation of the linear interpolation for E<sup>x</sup> components at the center position (x0, y0) of the

where N<sup>0</sup> is the central electron density of the bunch and w is the half width of the bunch. The coordinate (x0, y0) represents the center of electron-bunch, and its movement (trajectory) is

2σ<sup>2</sup> ( )

<sup>∂</sup><sup>t</sup> ¼ �nee <sup>E</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � � <sup>þ</sup> <sup>v</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � � � <sup>B</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � � � � (18)

� 2 =c<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>v</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � � � � �

<sup>2</sup>lnð Þ<sup>2</sup> <sup>p</sup> (17)

<sup>q</sup> (19)

(16)

spatial charge distribution:

118 Metamaterials and Metasurfaces

electron-bunch.

updated by solving following equation of motion:

<sup>∂</sup><sup>P</sup> <sup>x</sup>0; <sup>y</sup>0; <sup>t</sup> � �

In Figure 1, the analyzed 2D system and definitions of dimensions of the graded grating are schematically shown. In Table 1, parameters used in this study are summarized. The graded grating was assumed to be consisted of Ag, and the Drude model was adopted in order to model the dispersion characters of its dielectric function and solved by using RC scheme as discussed above. The plasma frequency (ωp) and the collision frequency (Γ) of Ag were set to

Figure 4. Schematic representation of simplified PIC-FDTD simulation scheme.

A 20-μm-wide (w) bunched electron-bunch with Gaussian charge distribution (Eqs. (16) and (17)) was sent 20 μm (w) above the grating at the relativistic speed. The maximum current

acceleration energy of the electron-bunch is 30 keV, which is comparable to the recent experi-

In the e-beam-induced radiations from conventional periodic grating, there are two mechanisms. One is the so-called Smith-Purcell radiation emitted while the e-beam is passing over the grating. The radiation angle and its frequency satisfy Eq. (1). The other is the scattering of surface waves at both ends of the grating long after the e-beam moved away from the grating. These long-lived surface waves can propagate back and forth on the grating surface and can be emitted repeatedly even long after the e-beam has moved away from the grating as long as the surface waves can live. The frequency of the scattered surface waves is determined by the intersection of the dispersion curves of the surface wave and the beam line. Here, we are interested in the second mechanism long after the e-beam has moved away from the gratings, but the groove depths are gradually graded, and, therefore, the dispersion curves of the

Figure 6(a) and (b) shows snapshot contour maps of the H<sup>z</sup> field long after the e-beam has moved away from the graded grating of GG[100, 168, 2] and GG[100, 168, 2], respectively. In both cases, directional radiations are obtained only from the shallowest end of the graded grating, in backward direction from GG[100, 168, 2] and forward direction from GG[100, 168, 2], respectively. These directional radiation characteristics cannot be expected from a conventional

surface waves induced by an e-beam cannot be uniquely determined.

periodic grating and might be unique in the graded grating considered here.

, and the

121

density at the center of the electron-bunch was assumed to be 1.0 106 A/m<sup>2</sup>

Grating period (Λ) 170 μm Groove width (s) 60 μm

Number of grooves (N) 35

Groove depth (d) Variable parameter

Electron Beam-Induced Directional Terahertz Radiation from Metamaterials

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Plasma frequency of Ag (ωp) 2.2 <sup>10</sup><sup>3</sup> THz Collision frequency of Ag (Γ) 5.4 THz Electron-bunch energy 30 keV Half width of electron-bunch (w) 20 μm Bunch-grating distance (w) 20 μm

mental condition by Urata et al. [5].

Table 1. Parameters used in this study.

3. Results and discussions

Figure 5. Flowchart of our PIC-FDTD scheme.

be 2.2 103 and 5.4 THz, respectively. The grating period (Λ) and the groove width (s) are 170 and 60 μm, respectively. The number of grooves of the grating (N) is 35. The groove depth (d) is gradually made deeper or shallower. Here, we denote each graded grating using grating parameters as GG[ds, dd, Δd], where d<sup>s</sup> and d<sup>d</sup> are the shallowest and the deepest groove depths, respectively, and Δd is the difference in depths between two consecutive grooves (all values are in μm). The space increment for the FDTD grids chosen here is Δx = Δy = 10 μm. The time increment is set to be Δt = 23 fs, which is sufficiently small to satisfy the condition for the stabilization of the FDTD algorithm.


Table 1. Parameters used in this study.

A 20-μm-wide (w) bunched electron-bunch with Gaussian charge distribution (Eqs. (16) and (17)) was sent 20 μm (w) above the grating at the relativistic speed. The maximum current density at the center of the electron-bunch was assumed to be 1.0 106 A/m<sup>2</sup> , and the acceleration energy of the electron-bunch is 30 keV, which is comparable to the recent experimental condition by Urata et al. [5].

### 3. Results and discussions

be 2.2 103 and 5.4 THz, respectively. The grating period (Λ) and the groove width (s) are 170 and 60 μm, respectively. The number of grooves of the grating (N) is 35. The groove depth (d) is gradually made deeper or shallower. Here, we denote each graded grating using grating parameters as GG[ds, dd, Δd], where d<sup>s</sup> and d<sup>d</sup> are the shallowest and the deepest groove depths, respectively, and Δd is the difference in depths between two consecutive grooves (all values are in μm). The space increment for the FDTD grids chosen here is Δx = Δy = 10 μm. The time increment is set to be Δt = 23 fs, which is sufficiently small to satisfy the condition for the

stabilization of the FDTD algorithm.

Figure 5. Flowchart of our PIC-FDTD scheme.

120 Metamaterials and Metasurfaces

In the e-beam-induced radiations from conventional periodic grating, there are two mechanisms. One is the so-called Smith-Purcell radiation emitted while the e-beam is passing over the grating. The radiation angle and its frequency satisfy Eq. (1). The other is the scattering of surface waves at both ends of the grating long after the e-beam moved away from the grating. These long-lived surface waves can propagate back and forth on the grating surface and can be emitted repeatedly even long after the e-beam has moved away from the grating as long as the surface waves can live. The frequency of the scattered surface waves is determined by the intersection of the dispersion curves of the surface wave and the beam line. Here, we are interested in the second mechanism long after the e-beam has moved away from the gratings, but the groove depths are gradually graded, and, therefore, the dispersion curves of the surface waves induced by an e-beam cannot be uniquely determined.

Figure 6(a) and (b) shows snapshot contour maps of the H<sup>z</sup> field long after the e-beam has moved away from the graded grating of GG[100, 168, 2] and GG[100, 168, 2], respectively. In both cases, directional radiations are obtained only from the shallowest end of the graded grating, in backward direction from GG[100, 168, 2] and forward direction from GG[100, 168, 2], respectively. These directional radiation characteristics cannot be expected from a conventional periodic grating and might be unique in the graded grating considered here.

Figure 6. Snapshot contour map of H<sup>z</sup> field long after the e-beam moved over (a) GG[100, 168, 2] and (b) GG[100, 168, 2].

As discussed by Gan et al. [16], the dispersion relations of the surface waves on these graded gratings are different at each location on the grating, and thus the frequencies of the e-beaminduced surface waves should also be different at different locations. These surface modes with different frequency components originated from different locations on the graded gratings can propagate toward the side with shallower groove depth due to the cutoff nature as reported by Gan et al. [16], which may give a mechanism of the directional radiation obtained only from the shallow end of the graded grating.

Figure 7(a) shows the time-domain H<sup>z</sup> field amplitude from GG[100, 168, 2] monitored at probe P indicated in Figure 6(a). After around 400 ps, successive pulse train with exponentially decaying magnitude can be seen. Magnified figure in the inset of Figure 7(a) indicates that each pulse train has duration of tens of ps and seems to be composed by the beating between several frequency components. Figure 7(b) shows the Fourier-transformed spectra of the far-field radiation from GG[100, 168, 2] monitored at probe P and from GG[100, 168, 2] monitored at probe Q. Both spectra have relatively wideband spectra and multiple sharp peaks, which cannot be expected from a conventional SPR in a periodic grating. The beating response seen in

time-domain response in Figure 7(a) should be attributed to these multiple sharp peaks. These two spectra from GG[100, 168, 2] and GG[100, 168, 2] are quite similar. The geometric parameters of these two graded gratings are identical except the groove depth variation Δd is opposite in sign; therefore, e-beam-induced surface modes are almost identical in both graded gratings, and only radiation direction was switched by making groove depth variation Δd opposite.

Figure 7. (a) Time-domain H<sup>z</sup> field amplitude from GG[100, 168, 2] monitored at probe P. (b) Fourier-transformed spectra of the far-field radiation from GG[100, 168, 2] monitored at probe P (red solid line) and from GG[100, 168, 2] monitored

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at probe Q (blue-dotted line).

In order to clarify the nature of the surface modes, we have also investigated near fields on the different locations on the graded gratings. Figure 8 shows Fourier-transformed

Figure 7. (a) Time-domain H<sup>z</sup> field amplitude from GG[100, 168, 2] monitored at probe P. (b) Fourier-transformed spectra of the far-field radiation from GG[100, 168, 2] monitored at probe P (red solid line) and from GG[100, 168, 2] monitored at probe Q (blue-dotted line).

As discussed by Gan et al. [16], the dispersion relations of the surface waves on these graded gratings are different at each location on the grating, and thus the frequencies of the e-beaminduced surface waves should also be different at different locations. These surface modes with different frequency components originated from different locations on the graded gratings can propagate toward the side with shallower groove depth due to the cutoff nature as reported by Gan et al. [16], which may give a mechanism of the directional radiation obtained

Figure 6. Snapshot contour map of H<sup>z</sup> field long after the e-beam moved over (a) GG[100, 168, 2] and (b) GG[100, 168, 2].

Figure 7(a) shows the time-domain H<sup>z</sup> field amplitude from GG[100, 168, 2] monitored at probe P indicated in Figure 6(a). After around 400 ps, successive pulse train with exponentially decaying magnitude can be seen. Magnified figure in the inset of Figure 7(a) indicates that each pulse train has duration of tens of ps and seems to be composed by the beating between several frequency components. Figure 7(b) shows the Fourier-transformed spectra of the far-field radiation from GG[100, 168, 2] monitored at probe P and from GG[100, 168, 2] monitored at probe Q. Both spectra have relatively wideband spectra and multiple sharp peaks, which cannot be expected from a conventional SPR in a periodic grating. The beating response seen in

only from the shallow end of the graded grating.

122 Metamaterials and Metasurfaces

time-domain response in Figure 7(a) should be attributed to these multiple sharp peaks. These two spectra from GG[100, 168, 2] and GG[100, 168, 2] are quite similar. The geometric parameters of these two graded gratings are identical except the groove depth variation Δd is opposite in sign; therefore, e-beam-induced surface modes are almost identical in both graded gratings, and only radiation direction was switched by making groove depth variation Δd opposite.

In order to clarify the nature of the surface modes, we have also investigated near fields on the different locations on the graded gratings. Figure 8 shows Fourier-transformed

Figure 8. Fourier-transformed spectra of near-field (surface wave) H<sup>z</sup> monitored at several positions 10 μm above each groove with depth of d = 160, 150, 140, 130, 120, 110, or 100 μm, along with that of the far-field radiation monitored at the probe P in GG[100, 168, 2] (from top to bottom). Markers A, B, and C refer to the peaks at 0.314, 0.329, and 0.347 THz, respectively.

In order to reveal from where each mode originate in the graded grating, we excited the system with quasi-monochromatic EM pulse and monitored long enough until the initial pulse damped and only long-lived surface modes survive. Figure 9 shows spatial distributions of H<sup>z</sup> fields for quasi-monochromatic long-lived surface modes of 0.314 (A), 0.329 (B), and 0.347 (C) THz. It can be seen that each surface modes originate at different locations of the graded grating and that higher-frequency modes originate at shallower grooves. This also supports that a superposition of the surface modes with different frequencies originated at different locations on the graded grating results in the directional and wideband far-field radiation with

Figure 9. Snapshot contour map of H<sup>z</sup> fields long after the exciting quasi-monochromatic electromagnetic pulse has been damped when the frequencies of the mode are 0.314 (A), 0.329 (B), and 0.347 (C) THz in GG[100, 168, 2]. Each mode and

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its name (A, B, C) correspond to the peaks in the far-field radiation spectrum in Figure 8.

The fact that the dispersion characters and frequency of the e-beam-induced surface mode is quite sensitive to the local environment of the grooves suggests that one can design the radiation frequency of the directional radiation from graded gratings by appropriately choosing groove parameters of the grating. Figure 10 shows the Fourier-transformed spectra of the far-field radiation from graded gratings with different groove parameters: GG[100, 304, 6], GG[100, 236, 4], GG[100, 168, 2], GG[168, 236, 2], GG[50, 186, 4], and GG[50, 118, 2]. Roughly speaking, the deepest and shallowest grooves determine the lowest and highest frequencies of the radiation, respectively. This can be confirmed by comparing spectra for GG[100, 304, 6], GG[100, 236, 4], and GG[100, 168, 2], for example. The highest frequency of these radiations is almost the same 0.40 THz and determined by their common shallowest groove depth d<sup>s</sup> of 100 μm. The concept of spoof SPPs may be promising for developing THz

multiple sharp peaks.

radiation sources.

spectra of the near-field (surface wave) H<sup>z</sup> field monitored just above each groove with d = 160, 150, 140, 130, 120, 110, or 100 μm, along with that of the far-field radiation monitored at the probe P, all in GG[100, 168, 2]. It can be seen that more peaks appear on the higher-frequency side of the spectra for monitoring above the shallower grooves. This is because the frequency of the surface waves and their cutoff frequency are lower at deeper grooves, and more surface modes can be supported above the shallower grooves. The spectrum monitored at the left end groove of the grating (d = 100 μm) is almost identical to the far-field radiation spectrum, which also supports that the directional and wideband far-field radiation with multiple sharp peaks was obtained as a superposition of all of the surface modes with different frequencies originated at different locations on the graded grating.

Figure 9. Snapshot contour map of H<sup>z</sup> fields long after the exciting quasi-monochromatic electromagnetic pulse has been damped when the frequencies of the mode are 0.314 (A), 0.329 (B), and 0.347 (C) THz in GG[100, 168, 2]. Each mode and its name (A, B, C) correspond to the peaks in the far-field radiation spectrum in Figure 8.

In order to reveal from where each mode originate in the graded grating, we excited the system with quasi-monochromatic EM pulse and monitored long enough until the initial pulse damped and only long-lived surface modes survive. Figure 9 shows spatial distributions of H<sup>z</sup> fields for quasi-monochromatic long-lived surface modes of 0.314 (A), 0.329 (B), and 0.347 (C) THz. It can be seen that each surface modes originate at different locations of the graded grating and that higher-frequency modes originate at shallower grooves. This also supports that a superposition of the surface modes with different frequencies originated at different locations on the graded grating results in the directional and wideband far-field radiation with multiple sharp peaks.

The fact that the dispersion characters and frequency of the e-beam-induced surface mode is quite sensitive to the local environment of the grooves suggests that one can design the radiation frequency of the directional radiation from graded gratings by appropriately choosing groove parameters of the grating. Figure 10 shows the Fourier-transformed spectra of the far-field radiation from graded gratings with different groove parameters: GG[100, 304, 6], GG[100, 236, 4], GG[100, 168, 2], GG[168, 236, 2], GG[50, 186, 4], and GG[50, 118, 2]. Roughly speaking, the deepest and shallowest grooves determine the lowest and highest frequencies of the radiation, respectively. This can be confirmed by comparing spectra for GG[100, 304, 6], GG[100, 236, 4], and GG[100, 168, 2], for example. The highest frequency of these radiations is almost the same 0.40 THz and determined by their common shallowest groove depth d<sup>s</sup> of 100 μm. The concept of spoof SPPs may be promising for developing THz radiation sources.

spectra of the near-field (surface wave) H<sup>z</sup> field monitored just above each groove with d = 160, 150, 140, 130, 120, 110, or 100 μm, along with that of the far-field radiation monitored at the probe P, all in GG[100, 168, 2]. It can be seen that more peaks appear on the higher-frequency side of the spectra for monitoring above the shallower grooves. This is because the frequency of the surface waves and their cutoff frequency are lower at deeper grooves, and more surface modes can be supported above the shallower grooves. The spectrum monitored at the left end groove of the grating (d = 100 μm) is almost identical to the far-field radiation spectrum, which also supports that the directional and wideband far-field radiation with multiple sharp peaks was obtained as a superposition of all of the surface modes with different frequencies originated at different locations on

Figure 8. Fourier-transformed spectra of near-field (surface wave) H<sup>z</sup> monitored at several positions 10 μm above each groove with depth of d = 160, 150, 140, 130, 120, 110, or 100 μm, along with that of the far-field radiation monitored at the probe P in GG[100, 168, 2] (from top to bottom). Markers A, B, and C refer to the peaks at 0.314, 0.329, and 0.347 THz,

the graded grating.

respectively.

124 Metamaterials and Metasurfaces

side of the grating with shallower grooves. The direction of these radiations can be switched backward or forward by making the groove depth deeper or shallower. The spectra of these directional radiations are wideband and contain multiple sharp peaks. The deepest and the shallowest groove depths determine the lowest and the highest frequency of the radiation band, respectively. These unique radiation characteristics cannot be explained by the conventional Smith-Purcell radiation and should be attributed to the spoof SPP that originates from different locations on the graded grating. The unique e-beam-induced radiation from metamaterials based on spoof SPP's concept may open a way for a development of novel types

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This work is partly supported from Okasan-Kato Foundation. The presented works have been carried out with graduate students who formerly belonged and currently belong to our

Department of Electrical and Electronic Engineering, Graduate School of Engineering,

[1] Tonouchi M. Cutting-edge terahertz technology. Nature Photonics. 2007;1:97-105. DOI:

[2] Hangyo M. Development and future prospects of terahertz technology. Japanese Journal

[3] Matsui T. A brief review on metamaterial-based vacuum electronics for terahertz and microwave science and technology. Journal of Infrared, Millimeter and Terahertz Waves.

of Applied Physics. 2015;54:120101. DOI: 10.7567/JJAP.54.120101

2017;38:1140-1161. DOI: 10.1007/s10762-017-0387-9

of THz radiation sources.

Acknowledgments

Conflict of interest

Author details

Tatsunosuke Matsui

References

Mie University, Tsu, Japan

10.1038/nphoton.2007.3

There is no conflict of interest.

research group, Okajima, Omura, and Yoshida.

Address all correspondence to: matsui@elec.mie-u.ac.jp

Figure 10. Fourier-transformed spectra of the far-field radiation from graded gratings with different groove parameters: GG[100, 304, 6], GG[100, 236, 4], GG[100, 168, 2], GG[168, 236, 2], GG[50, 186, 4], and GG[50, 118, 2] (from top to bottom).

## 4. Conclusions

We have numerically analyzed the e-beam-induced directional THz radiation from metallic grating structures with graded depths. We used a simplified PIC-FDTD method for numerical analysis to save computational time and memory, and the detailed description of our method is given here. In our simplified model, the electron-bunch is treated as one negatively charged particle with Gaussian charge distribution, and its movement is restricted only in 2D space, and only TE mode, with Ex, Ey, and H<sup>z</sup> fields, has been analyzed.

Our results show unique directional THz radiation from graded gratings. By passing pulsed (bunched) e-beam along the grating surface, directional THz radiations are obtained from one side of the grating with shallower grooves. The direction of these radiations can be switched backward or forward by making the groove depth deeper or shallower. The spectra of these directional radiations are wideband and contain multiple sharp peaks. The deepest and the shallowest groove depths determine the lowest and the highest frequency of the radiation band, respectively. These unique radiation characteristics cannot be explained by the conventional Smith-Purcell radiation and should be attributed to the spoof SPP that originates from different locations on the graded grating. The unique e-beam-induced radiation from metamaterials based on spoof SPP's concept may open a way for a development of novel types of THz radiation sources.
