Characterization of Electrooptic Modulator
4.1 Microwave Analysis
In the analysis of optical modulators, the scalar potential function ( , )x y of the induced electric field can be determined over the cross section of the modulator by solving the two dimensional Laplace‟s equation;
2 2
2 2
( , ) ( , ) 0
x
x y
yx y
x y
, (4.1)where x and y are the permittivities of anisotropic LiNbO3 substrate in the x and y directions.
When FEM is used over the modulator cross section, the TEM matrix equation for Laplace‟s equation follows;
[ ]{ } {0}K
(4.2)with
e e 0(
rx
T ry
T)
N N N N
K dxdy
x x y y
, (4.3)where the Lagrange‟s triangular elements are used for discretizing the modulator cross section, {} is the vector for nodal potentials, {0} is a null vector, [K] is the stiffness matrix, 0 is the permittivity of free-space, rx and ry are the relative permittivities in x and y directions, respectively. Equation (4.2) can be reduced by using Dirichlet type known boundary conditions.
Discretizing the modulator cross section using adaptive meshing and solving the resultant system, the potential distribution can be determined. When the potential is known, both the horizontal and vertical electric fields, Ex and Ey, can be determined by using;
( , )x y
E , (4.4)
where E is the electric field vector and can be calculated using the nodal potentials as;
3 1
1 ˆ ˆ
2 A
e jb x c y
ej ej
ej
E
, (4.5)where b1 y2y3, b2 y3y1, b3 y1y2,
3 2
1 x x
c , c2 x3x1, c3 x1x2
The x‟s and y‟s are coordinates of nodes of each small triangle produced during the FEM meshing over the cross section of the modulator. The area of each small triangle is given by;
Ae =
3 3
2 2
1 1
1 1 1
y x
y x
y x
(4.6)
In FEMLAB the whole area except the electrodes of the modulator is divided into small triangular cells.
4.1.1 Calculation of Electrode Capacitance
The capacitance of the electrode, C, is calculated using the energy expression as [7], [22];
2
2 2
0 0
2 1 ( , )
S
C W x y dxdy
V V
, (4.7)where [] is the permittivity tensor for anisotropic materials, V0 is the potential difference between the electrodes, W is the total energy per unit length, s is the cross sectional domain of the structure. In the calculation however, we used the electric fields as;
2 2
o2
rx x ry y e
o e
C E E A
V
(4.8)where Ex and Ey are the x and y components of electric field vector E.
4.1.2 Calculation of Microwave Index and Characteristics Impedance
In calculating effective index of the microwave, Nm and characteristic impedance, Zc free space capacitance
C0 and electrode capacitance
C are required to be calculated. The capacitance of the CPW line C can be calculated by using equation (4.8). By replacing the dielectric materials by free space, capacitance of the free space line C0 can be calculated using the same equation. By using C and C0 the Nm and Zc can be calculated using;C0
Nm C (4.9)
0
1 CC
Zc c (4.10)
4.1.3 Calculation of Microwave Loss
In high speed modulator, when phase velocity matching is achieved, the limiting factor arises from overall propagation loss [13], [20]. The dominance of the loss components depends on frequency. At lower operating frequencies, the electrode‟s conductor loss dominates; whereas, when the operating frequency if increased beyond 40GHz, the dielectric loss is expected to play an increasingly important role in the determination of bandwidth [13], [20].
Conductor loss coefficient:
The conductor loss due to the imperfection of conductor can be calculated with the incremental inductance formula as;n Z Z Z
R n L Z
R cO
c o
s c
c s
2 2
, (4.11)
where ZcO is the characteristic impedance of the electrode in free space and n ZcO
denotes the derivative of ZcO with respect to the incremental recession (half of skin depth) of the electrode surface. Rs is the surface resistance and can be calculated by the following equation [22];
12
) ( f
Rs o , (4.12)
where f denotes modulating frequency and is the metal resistivity.
Dielectric loss coefficient:
The microwave loss that is due to lossy dielectrics can be calculated with the perturbation approach [13], [32] to give2P0
Pd
d
, (4.13)
wherePo is the time average power flow along the line, and Pd is the time average power dissipated in the dielectrics.Pd can be calculated using the formula [32]
dS E
pd tan
Sdiel 0 2 , (4.14)where represents the loss tangent of the dielectric region, 2f is the angular frequency, and Sdiel is the area of cross section covered by the dielectrics. Equation (4.14) can be written in the simplified form as,
E E
dSpd tan
Sdiel x2 y2 . (4.15)The average power of propagation along the line,Po, is given by Poynting Vector as [23]
S
o E H
P Re 0 0* .zdS, (4.16)
where S is the complete cross section of the modulator. Here, the electromagnetic field distribution can be obtained using (4.5) for unperturbed fields [32]
0
E , (4.17)
0 1 0 ˆ
H E z, (4.18)
where zˆ is the unit vector in the z-direction and is the intrinsic impedance of the medium as given by
12 0 0 r
, (4.19)
Equation (4.16) can be written into the following form (derivation is given in appendix A)
2 0
1
m
o N
P
S
Ex2 Ey2
dS (4.20)Here Ex and Ey are respectively the x axis and y axis components of the electric field, Nm is the microwave index, and 0 is the intrinsic impedance of the free space.
4.1.4 Calculation of Optical Response and 3-dB Bandwidth
The optical response of a modulator is determined by the microwave propagation characteristics of the electrode, namely, the effective index of the microwave,
N
m, the characteristic impedance,Z
C, and the overall propagation losses. The general equation of the optical response is given as [3], [10]
u
ju u u S
ju u ju
S S u j S
S f S
m exp( )sin exp( )sin
) 2 exp(
) 2 exp(
) 1 ( ) 1
( 2
2 1 2
2 1
(4.21)
where ,
2 ) 1
1 fL(N N j L
u c m o
c c
Z Z
Z S Z
1
1 1 and
c c
Z Z
Z S Z
2
2 2 . (4.22)
Here Zc, Z1, and Z2 are the microwave characteristic impedance, the microwave generator's internal impedance, and the shunted loaded impedance, respectively. Here Nm, is the microwave effective index, N0 is the optical effective index, represents overall microwave and dielectric losses, and c is the velocity of light [20]. Under the impedance-matching condition, ZC = Z1 =Z2, which is often set to 50Ω, the bandwidth is limited by the velocity mismatch and the total microwave loss, and thus the optical response equation is reduced to
u
ju u f
m( ) exp( )sin . (4.23)
Equation (4.21) can be simplified to the following form (derivation is shown in Appendix A).
12 2 2
2
) 2 ( ) (
2 cos 2
) 1
(
u L
e u f e
m L L
(4.24)
where 1 ( )
o
m N
N c fL
u (4.25)
and c f (4.26)
Under both impedance and phase-matched conditions, ZC = Z1 = Z2 and Nm = No, where No is often taken as 2.3, the bandwidth is limited by the microwave loss only, the optical response equation is further reduced with the 3-dB optical bandwidth, Δf, and is given by [13].
12
84 . 6
f L
c , (4.27)
where c is the conductor loss in decibels per unit length normalized at 1GHz. However, in the estimation of the modulator bandwidth, the critical factor is the velocity mismatch, and, when this is not achieved, the 3-dB optical bandwidth Δf is determined approximately by [13]
L N N f c
m
0
2
. (4.28)
In the calculation of optical bandwidth, often only the conductor loss is considered. However, when the dielectric loss is included, then the total frequency dependent loss may be given by [13]
f f
f c d
( ) . (4.29)
If cand d is normalized at 1 GHz, then f is given in GHz. Herec is the conductor loss in decibels per (square root of gigahertz * centimeters), and d is the total dielectric loss in units of decibels per (gigahertz * centimeters). The dielectric loss can arise from different lossy microwave regions, and in this case
dclad dSpac
dCore dB
dSub
d
, (4.30)
where dSub, dB, dCore, dSpac and dClad, are the dielectric losses in the substrate, buffer layer, core region, spacer layer and cladding region, respectively.