Andrzej Szczepkowicz, Levi Schächter, and R. Joel England, "Frequency-domain calculation of Smith–Purcell radiation for metallic and dielectric gratings," Appl. Opt. 59, 11146-11155 (2020)
The intensity of Smith–Purcell radiation from metallic and dielectric gratings (silicon, silica) is compared in a frequency-domain simulation. The numerical model is discussed and verified with the Frank–Tamm formula for Cherenkov radiation. For 30 keV electrons, rectangular dielectric gratings are less efficient than their metallic counterparts, by an order of magnitude for silicon, and two orders of magnitude for silica. For all gratings studied, radiation intensity oscillates with grating tooth height due to electromagnetic resonances in the grating. 3D and 2D numerical models are compared.
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Cherenkov Radiation: Verification of the Numerical Results Against the Frank–Tamm Formula for One Frequency, a
Velocity and Refractive Index
,
1.04
,
1.08
,
1.04
,
No radiation
Here we compare the numerical value ${d^2}{W_{{\rm num}}}/dz d\omega$ against the analytical value ${d^2}{W_{{\rm analytical}}}/dz d\omega = ({e^2}/4\pi){\mu _0}\omega (1 - 1/{\beta ^2}{n^2})$ (see Appendix B); the second column shows the ratio of these two values.
Table 2.
SP Radiation from Gratings of Different Materialsa
Relative Permittivity ()
Grating Material
Real Part
Imaginary Part
Radiated Energy
Energy Radiated into the Grating
Copper
1.361
0
Gold
1.462
0
Perfect conductor
0
0
Fused silica
2.107
0
Silicon
13.32
0.03099
Emitted perpendicular to the grating within the frequency range $2\pi \cdot 325.5 \; {\rm THz} \lt \omega \lt 2\pi \cdot 330.5\;{\rm THz} $, corresponding to angular range ${88.7^ \circ} \lt \theta { \lt 91.3^ \circ}$, per electron per grating period, for the grating geometry and beam parameters from Figs. 1(a)–1(b). Relative permittivity is taken from Refs. [58–61].
Table 3.
Comparison of 3D and 2D SP Radiation in the Angular Range for a Metallic Grating with Beam/Grating Configuration in Fig. 1
Geometry of the Model
Radiated Energy
3D model
2D model,
2D model,
Table 4.
SP Radiation from Metallic Triangular Gratings, in the Angular Range , per Electron per Grating Period, for Grating Geometry Shown in Fig. 7a
Grating Tooth Height
, Numerical Model
, Analytical Model
Ratio
40 nm
4.0
80 nm
3.5
160 nm
2.4
320 nm
1.1
We compare our numerical results with the analytical model in Ref. [28].
Tables (4)
Table 1.
Cherenkov Radiation: Verification of the Numerical Results Against the Frank–Tamm Formula for One Frequency, a
Velocity and Refractive Index
,
1.04
,
1.08
,
1.04
,
No radiation
Here we compare the numerical value ${d^2}{W_{{\rm num}}}/dz d\omega$ against the analytical value ${d^2}{W_{{\rm analytical}}}/dz d\omega = ({e^2}/4\pi){\mu _0}\omega (1 - 1/{\beta ^2}{n^2})$ (see Appendix B); the second column shows the ratio of these two values.
Table 2.
SP Radiation from Gratings of Different Materialsa
Relative Permittivity ()
Grating Material
Real Part
Imaginary Part
Radiated Energy
Energy Radiated into the Grating
Copper
1.361
0
Gold
1.462
0
Perfect conductor
0
0
Fused silica
2.107
0
Silicon
13.32
0.03099
Emitted perpendicular to the grating within the frequency range $2\pi \cdot 325.5 \; {\rm THz} \lt \omega \lt 2\pi \cdot 330.5\;{\rm THz} $, corresponding to angular range ${88.7^ \circ} \lt \theta { \lt 91.3^ \circ}$, per electron per grating period, for the grating geometry and beam parameters from Figs. 1(a)–1(b). Relative permittivity is taken from Refs. [58–61].
Table 3.
Comparison of 3D and 2D SP Radiation in the Angular Range for a Metallic Grating with Beam/Grating Configuration in Fig. 1
Geometry of the Model
Radiated Energy
3D model
2D model,
2D model,
Table 4.
SP Radiation from Metallic Triangular Gratings, in the Angular Range , per Electron per Grating Period, for Grating Geometry Shown in Fig. 7a
Grating Tooth Height
, Numerical Model
, Analytical Model
Ratio
40 nm
4.0
80 nm
3.5
160 nm
2.4
320 nm
1.1
We compare our numerical results with the analytical model in Ref. [28].
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