High speed, low driving voltage vertical cavity germanium-silicon modulators for optical

5.1.3.1 Basic Model 
The two port system shown in Figure 5.4 can be characterized by Z (impedance) 
parameters at low frequencies. The equations for the impedance parameters can be 
written as: 

---

 
 
 
71 
1
1
1
1
1
2
2
V
Z
I
Z
I


(5.1) 
2
21 1
22 2
V
Z I
Z I


(5.2) 
Figure 5.4: Two-port system model: variable definitions 
Open-circuit ports need to be used to determine the Z parameters experimentally, 
For example, Z
11
is determined easily once output port 2 is open-circuited (I
2
=0), and 
measuring voltage V
1
and current I
1
allows direct computation of Z
11
, which is 
2
1
11
1
0
I
V
Z
I


(5.3) 
At high frequencies, the approach mentioned above does not work. Since it is 
impossible to have open or short circuits, a different method needs to be used. Another 
set of parameters, called scattering parameters (or S-parameters), can fit into the high 
frequency scenario very well. They relate the incident and reflected voltage waves 
rather than port voltages and currents, as shown in Figure 5.5. The source and load 
terminations are Z
0
. The high frequency two port relation using S-parameters can be 
expressed as 
1
11
1
12
2
r
i
i
E
s E
s E


(5.4) 
2
21
1
22
2
r
i
i
E
s E
s E


(5.5) 

---

 
 
 
72 
Figure 5.5: High frequency two-port system model 
The S parameters can be found by driving one port and measuring at the other port. 
They can be determined from the following equations: 
1
11
1
1
r
i
E
s
E

 
(5.6) 
2
21
1
r
i
E
s
E

(5.7) 
S
11
is the input reflection coefficient and S
21
is the gain relating the output 
reflected wave to the input wave. S
12
and S
22
can be deduced in a similar way. Also, 
the reflection can be written as
1
0
1
0
Z
Z
Z
Z

 

(5.8) 
where Z
1
is the load impedance. With this model, a more intuitive picture can be 
used to model the behavior of the device.