Supplement material for "Coherent and Tunable Terahertz Radiation from
Graphene Surface Plasmon Polaritons Excited by an Electron Beam"
Appendix I : Graphene with dielectric substrate
Figure S1. The schematic diagram of a graphene sheet on a substrate, region I is
vacuum and region II is the substrate (  s ).
As shown in Fig.S1, a monolayer graphene is on a dielectric substrate, and an electron beam is
uniformly moving parallel above the graphene. Here MKS unit system is used. We consider the
TM mode wave in graphene, and then by making use of the Maxwell’s equations, the fields in
region I and region II can be written as:
Ez I  A1a e jk1 y e jk z z
Ez II  A2b e jk2 y e jkz z
where k z
k1 
 k0 
H xI 
 0
k1
 0 s
H x II  
k2
A1a e jk1 y e jk z z
A2b e  jk2 y e jk z z
(I.1)
(I.2)
is the propagation constant of graphene surface plasmon polaritons (SPPs),
2
 k z 2 , k2   s  k0   kz 2 and c is velocity of light in vacuum,  0 is the
2
dielectric constant in vacuum, k0 is
 c , and  s is the relative dielectric constant in region II.
 j t
The factor e
is neglected.
Assuming that monolayer graphene is atomically thin, we consider it as a conductive surface
with conductivity  g , then the boundary conditions are shown as below:
Ez I

ey  H x I
y 0
y 0
 Ez II
 H x II
(I.3)
y 0
y 0
  E
g
I
z
y 0
(I.4)
Substituting Eq. (I.1) and (I.2) into the above boundary conditions, the dispersion equation of
SPPs in monolayer graphene is obtained.
1
1
k02  k z 2
s

 s k02  k z 2
g
0
 0

(I.5)
Based on Eq. (I.5), Fig.1 (b) in manuscript is obtained. In the case of k0<<kz, and neglect the
second summand in Eq. 1 in the manuscript, the dispersion relation will take a very simple form,
whereby the angular frequency is proportional to the square root of the in-plane wave vector.
By making use of the Wronskian approach, the incident waves of the parallel moving electron
bunch are obtained.
Ez  y, z ,   
kz q
jk y  y
1   2  e jkz z e c 0

2 0u0 kc
(I.6)
k02  k z 2 , kz   u0 , u0 is the velocity of the moving electron bunch, q is
where kc 
the charge quantity of the bunch,
  u0 c , and y0 is the position of the bunch. The
description of fields in each region is the same as in the derivation of dispersion equations except
that k z   vSPPs is replaced by k z   u0 , where vSPPs is the phase velocity of SPPs.
The boundary conditions can be written as
Ez I
y 0
 Ez i
y 0
 Ez II
y 0

 H xI
y 0
 H xi
y 0
 H
II
x
y 0
  Ez II
y 0
(I.7)
And then the coefficient of the electric field in region II can be found as
A2b  jB i
where B 
i
 0 kc k2   0 k1k2
 0 kc k2   0 s k1kc   k1k2 kc
kz q
1   2  e jkc y0 .
2 0u0 kc
The contour map in Fig.1 (c) in the manuscript is calculated from this equation.
Appendix II. Graphene with dielectric micrometer slits array
Figure S2. Schematic diagram of a monolayer graphene with micrometer dielectric
slits array substrate, the slits array is formed by dielectric medium 1 and medium 2
alternatively, the height of the slits array is d, the widths of medium 1 and medium 2
2
(I.8)
are a1 and a2, an electron bunch is uniformly moving above the slits array at a
distance y0 .
2.1 Dispersion equation
As shown in Fig. S2, the scheme can be divided into three regions. By making use of the
Maxwell’s equations, we can get the following homogeneous Helmholtz equation. (In the
Cartesian coordinate shown in Fig. S2.)
 2 Ez  2 E z  2 E z
 2 Ez




0
x 2
y 2
z 2
t 2
(II.1)
Solving Eq. (II.1) together with the boundary conditions, the E z component of the fields in
region I and III can be obtained. And then all the other field components can be obtained by
Maxwell’s equations. The factor e
 j t
is neglected.
In the region I ( y  0 ):
EzI  B1e jk1 y e jk z z，H xI 
where
 0
k1
B1e jk1 y e jk z z
(II.2)
k1  k02  kz2 , k0   c , k z   v p , v p is the phase velocity of SPPs.
In the region III ( y  d ):
EzIII  B2e  jk3 y e jkz z , H xIII  
where k3 
 0 s
k3
B2e  jk3 y e jk z z
(II.3)
 s k02  kz2
In the region II (  H  y  0 ):
The dielectric slits array can be treated as a homogeneous dielectric layer with effective relative
permittivity  eff   s (1  r )   2 r , r  a2 / L , using the same method described in above, the
fields in region II can be written as:
EzII  ( B3e jk2 y  B 4e jk 2y )e jkz z ,H xII 
where k2 
 0 eff
k3
B( e3 jk y2  B e4 jk y 2 e )jk z z
(II.4)
 eff k02  k z2
The monolayer graphene is also considered as a conductive surface with conductivity
3
g ,
using the similar boundary conditions as presented in appendix I, the dispersion relation can be
obtained:
 eff k3   s k2 2 jk d  eff k0 k1  0 g k1k2  k0 k2
e

 eff k3   s k2
 eff k0 k1  0 g k1k2  k0 k2
2
(II.5)
where 0 is the wave impendence in vacuum.
In analog to translational invariance of crystal lattice, the periodicity of the dielectric media
results in zone folding of the SPP dispersion, so Fig. 2(b) in manuscript is obtained.
2.2 Power density
By making use of Maxwell’s equation, we can get the following nonhomogeneous Helmholtz
equation.
 2 Ezi  2 Ezi  2 Ezi 1  2 Ezi
J
 2  2  2
 z
2
2
x
y
z
c t
t
J z  q u0  
y
y   z
u
0
0
t
(II.6)
(II.7)
Solving the nonhomogeneous Helmholtz equation (II.6), the E zi component of source field
induced by electron beam can be obtained.
Ezi  
 q
2kc
(1 
1

2
)e
jkc y  y0
e jkz z
(II.8)
k02  k z2 , kz   u0 ,   u0 c , u0 is the velocity of moving electron bunch.
where kc 
To obtain the radiation power density, the fields in each region should be expanded to space
harmonic waves.
In the region I ( y  0 ):
EzI   A1n e jk1n y e jk zn z，H xI  
n
where k zn  k z 
n
 0
k1n
A1n e jk1n y e jk zn z
(II.9)
A2 n e  jk3 n y e jk zn z
(II.10)
2 n
2
2
, k1n  k0  kzn , k0   c .
D
In the region III ( y  d ):
EzIII   A2 n e  jk3 n y e jkzn z , H xIII   
n
where k3n 
n
1k02  kzn2 .
4
 0 s
k3 n
The periodic relative permittivity is expandable in a Fourier series of the form:
 r ( z )   c p exp(i
p
2 p
)
L
(II.11)
where the coefficients are:
 s (1  r )   2 r

cp  
sin[ p(1  r )]
( s   2 )
p

p0
p0
, r  a2 / L
and its inverse is :
1
2 p
   p exp(i
)
 r ( z) p
L
(II.12)
where the coefficients are:
1
1
  (1  r )   r

2
p   s
( 1  1 ) sin[ p(1  r )]
  s  2
p
p0
p0
The fields in region II can be written as:
EzII   S zn ( y)e jkzn z , H xII  j
n
0
U xn ( y)e jk

0 n
zn z
(II.13)
where U x , S z satisfy the eigen equation of dielectric slits array and their space harmonics are
then given by
U x  {bi exp[qi ( y  d )]  bi exp(qi y )}wi
(II.14)
i
S z  {bi exp[qi ( y  d )]  bi exp(qi y)}vi ,
(II.15)
i
where wm ,i and qi are the elements of the eigenvector matrix W and the positive square root
1
of the eigenvalues of the matrix A B , B  K z CK z  k0 I , C and A are the matrix formed
2
by the permittivity and inverse permittivity components, K z is a diagonal matrix, with the i , i
element being equal to k zi , I is the identity matrix. The matrix V 
5
1
AWQ , Q is a
k0

diagonal matrix with the elements qi . The quantities bi

and bi
are unknown constants to be
determined from boundary conditions.
Matching the boundary conditions, the coefficients of fields in all regions can be obtained. The
radiation power per unit area can then be obtained by:
P
 A2
1
Re[ 0 1n ]
2
k1n
n
(II.16)
where
A1  VXb   Vb   M 3 , A2  M 11 (Vb   VXb  ) ,
b   [(WX  M 5VX )( M 1W  M 2V ) 1 ( M 2VX  M 1WX )  ( M 5V  W )]1 ( M 4  M 5 M 3 )
b   ( M 1W  M 2V )1 ( M 2VX  M 1WX )b  ,
M 1 , M 2 and M 5 are diagonal matrices with the i , i elements being equal to e jk3i d ,
jk
0
j s k0 jk3id
and  0  j
e
 , M 3 and M 4 are column matrix with zero elements
k3 i
k1i
0 g
except the middle elements being equal to
jk
0
 q jkc y0
 q jkc y0
, ( 0j
g)
e .
e
2kc
kc
0
2 kc
The contour map in Fig.2 (c) and the spectrum of E z component of radiation field in the
manuscript are calculated.
2.3 Radiation characteristic dependence on the height of dielectric slits array
The radiation characteristic may be influenced by the height of dielectric slits array. We study
the radiation frequency and power density dependence on the height varying from 0.6 to 1.6 m.
Due to the high confine of graphene SPPs on graphene-dielectric surface, its dispersion curve is
almost unaffected by the substrate below the dielectric slits array when the height is larger than 0.6
um, so the radiation frequency is nearly unchanged. When the height is smaller than 0.6 m, the
substrate will affect the dispersion of graphene SPPs. The Fig.S3 shows the dependence of
radiation power density on the height of dielectric slits array. The power density increases with the
increasing height, reaches to a peak, and then decreases. To ensure the high radiation power
density and the independence of radiation frequency on substrate below the dielectric slits array,
the height of dielectric slits array is set to be comparable to the length of the period.
6
Figure S3.Radiation power density dependence on the height of dielectric slits array
2.4 The radiation power density dependence on various different parameters
Fig.S4 shows the radiation power density dependence on various different parameters. When
each of the parameters (chemical potential, ratio of slit width to period, period and relative
permittivity) is fixed, the optimization radiation power density is obtained by choosing appropriate
beam energy. For example, in Fig.S4 (a), if the chemical potential is 0.15 eV, the max radiation
power density is about 100 mW/cm2.. And the radiation power density decreases with the
decreasing chemical potential.
7
Figure.S4.Radiation power density dependence on various different parameters (a)
Radiation power density vs. chemical potential (b) Radiation power density vs. ratio
of slit width to period (c) Radiation power density vs. period (d) Radiation power
density vs. relative permittivity, the beam energies (β) are labeled in the brackets.
Appendix III: Graphene with PEC micrometer slits array
As shown in Fig.S3, a PEC micrometer slits array replaces the dielectric substrate, and it is a
two dimensional periodic structure.
Figure S5. Schematic diagram of graphene in PEC micrometer slits array Here MKS
unit system is used, and we consider the TM mode in graphene. By making use of the
Maxwell’s equation, the fields in each region can be written.
For region I:


Ez I   A1n a e jk1n y e jkzn z
n
 

H x I    0 A1n a e jk1n y  e jkzn z
n  k1n

(III.1)
where k z is the propagation constant of graphene SPPs, k zn  k z  2 n D , n is the order of
 c 
space harmonics, k1n 
2
 kzn 2 , and c is velocity of light in vacuum,  0 is the
 j t
dielectric constant in vacuum. The factor e
is neglected.
For region II:


Ez II   A2 n a e jk2 n y  A2 n b e  jk2 n y e jk zn z
n
  

 
H x II    0 d A2 n a e jk2 n y  0 d A2 n b e  jk2 n y  e jk zn z
k2 n
n  k2 n

where k2n 
 b  c   kzn 2 ,  d is the relative dielectric constant in region II.
2
For region III:
8
(III.2)
Ez III
 a jk0 y
b  jk0 y
 A3 e  A3 e

0

a
a
z
2
2
a
a
 z  D
2
2

 0
A3a e jk0 y  A3b e  jk0 y

 0


0

Hx
III

(III.3)
a
a
 z
2
2
a
a
 z  D
2
2
where k0   c and 0 is the permeability in vacuum.
For region IV:

  

H x IV     0 s A4 nb e jk4 n y  e jkzn z
k4 n
n 


Ez IV   A4 nb e jk4 n y e jkzn z
n
where k4n 
(III.4)
 s  c   kzn 2 ,  s is the relative dielectric constant in region IV.
2
The monolayer graphene is considered as a conductive surface, and then the boundary
conditions for region I and II are written as:
Ez I
y  h1
 Ez II
y  h1

ey  H x I
y  h1
 H x II
y  h1
E
I
z
y  h1
(III.5)
And then matching the fields in each region, the dispersion equation of graphene SPPs in PEC
periodic structure can be obtained.
Q1 1Q2 1 e jk h  Q2 1Q1  1 e jk h
0 2
0 2
(III.6)
where
a
b
k0 d   M n  M n 
k a
a
Q1   
sin c 2 zn
a
b
D n k2 n  M n  M n 
2
M na 
Q2  
k0 s
k a
a
sin c 2 zn

D n k4 n
2
e jk1n h1  0 d k1n   0  k1n  k2 n
e jk1n h1  0 d k1n   0  k1n  k2 n
b
M

n
e jk2 n h1
2 0 d k1n
e jk2 n h1
2 0 d k1n
By making use of the Wronskian approach, the incident waves of the parallel moving electron
bunch are obtained.
Ez  y, z ,   
where kc 
kz q
1   2  e jkz z e jkc y  y0
2 0u0 kc
(III.7)
k02  k z 2 , kz   u0 , u0 is the velocity of the moving electron bunch, q is
the charge quantity of the bunch,
  u0 c , and y0 is the position of the bunch. The
description of fields in each region is the same as in the derivation of dispersion equations except
9
that k z   vSPPs is replaced by k z   u0 , where vSPPs is the phase velocity of SPPs.
Matching the fields in each region, the coefficients in region III can be obtained.
A3b 
 S2  S1  e jk h  Sum34  1
 Sum34  1 Sum23  1  e jk h  Sum34  1 Sum23  1
0 2
e jk0h2
(III.8)
0 2
A3a   A3b
e jk0h2  Sum34  1
e jk0h2  Sum34  1
(III.9)
where
Sum23  
n
S1   Bi
Pa 
k0 d
k a
k    M 0 a  M 0b  a
k a
 P a  Pb  sin c z 0 ,S2   Bi 0 d 
P  Pb  sin c z 0

a
b 
k20
2
k20  M 0  M 0 
2
e jkc h1   kc k20   0 d kc   0 k20  b e jkc h1   kc k20   0 d kc   0 k20 
, P   jk20h1
e jk20h1
2 0 d kc
e
2 0 d kc
M 0a 
M na 
Bi 
k a
k a
a k0 d  M n a  M nb 
a k 
sin c 2 zn , Sum34    0 s sin c 2 zn 

a
b 
D k2 n  M n  M n 
2
2 
n D  k4 n
e jk10 h1   k10 k20   0 d k10   0 k20 
e jk10 h1   k10 k20   0 d k10   0 k20 
b
,
M

0
e jk20 h1
2 0 d k10
e  jk20 h1
2 0 d k10
e jk1n h1   k1n k2 n   0 d k1n   0 k2 n 
e jk1n h1   k1n k2 n   0 d k1n   0 k2 n 
b
,
M

n
e jk2 n h1
2 0 d k1n
e  jk2 n h1
2 0 d k1n
kz q
1   2  e jkc y0

2 0u0 kc
Appendix IV: The mechanism of enhancement
Graphene SPPs are considered as surface current density: J zsp   g Ez , which is the
equivalent source for the enhancement. The total radiation power consists of attribution of J zsp
and electron beam without the graphene sheet. As shown in Appendix II, the fields excited by the
parallel electron beam in each region are obtained from the boundary conditions as below.
( Ezi  EzI ) | y 0  EzII | y 0
H xII | y 0 ( H xi  H xI ) | y 0   g ( EzII ) | y 0
EzII | y  d  EzIII | y  d
(IV. 1)
H xII | y  d  H xIII | y  d
And then the equivalent current is found as
 g ( EzII ) . Then we have the follow boundary
conditions for this structure excited by the equivalent current.
10
 E  |
I
z
y 0
 H  |
II
x
|
E 
II
z
II
where E z
y 0
  H xI  | y 0   g EzII | y 0
y  d
 H  |
II
x
  EzII  | y 0
y  d
 E
III
z
|

(IV. 2)
y  d
  H xIII  | y  d
is the filed excited directly be the parallel beam, and the fields with superscript  are
the fields excited by the equivalent current. The numerical calculation shows that the fields excited
by the equivalent current is much larger than the fields excited by the beam without the graphene
sheet, which agree well with the results given in Fig.4 in our manuscript.
11