# Review of CMOS Integrated Circuit Technologies for High-Speed Photo-Detection
## 5 Bandwidth Extension Techniques
### Equalization  
In the electrical link receiver, various types of equalization techniques have been utilized to overcome the frequency dependent loss of the electrical transmission line. On the other hand, there has been little interest in using such equalization techniques for optical links because optical fibers  
Figure 35: Effect of feedback on the transfer function of an amplifier (**a**) with only resistive feedback (**b**) with the combination of resistive and inductive feedback. provide almost infinite transmission bandwidth. However, equalization techniques have been adopted in recent state-of-the-art optical receivers in order to (1) overcome the limited O/E converting speed of PD, (2) break the tradeoff between high-gain and high-bandwidth, and (3) enhance the TIA bandwidth with minimal added noise. This section summarizes the equalization techniques for a photodetecting circuit that have been presented in leading journals and conferences recently. In 2010, a University of Toronto group proposed a TIA with negative Miller capacitances and a continuous-time linear equalizer (CTLE) [106] in order to overcome the limited speed of a spatially modulated PD. Figure 36 shows the principle of the Miller effect. In definition, the impedance is the coefficient of the voltage difference between the two ports of a device under test (DUT) over the current flowing through the DUT. In case there is an amplifier whose input and output are connected to the two ports of a DUT, the voltage applied to the DUT is influenced by the amplifier as well as the input voltage. Therefore, the amplifier gain should be included in the impedance. For example, the input impedance becomes \(Z/(1\)-\(A)\), where \(Z\) and \(A\) represent the original impedance of the DUT and gain of the amplifier, respectively. It can be achieved by using the Kirchhoff's current law (KCL) at the input node, as long as the amplifier sets the ratio of the output voltage over the input voltage to \(A\). Because the [PERSON] effect is a truly well-known circuit effect, further explanations can be found in many textbooks, i.e. [107]. In [106], a pair of capacitors is placed between the input and output of a differential CS stage TIA, which have the same polarity, as shown in Figure 36b. Owing to the [PERSON] effect, the input capacitance of the TIA becomes \(C_{CS}+C(1\)-\(A_{CS})\), where \(C_{CS}\), \(C\), and \(A_{CS}\) are the input capacitance of the CS stage, Miller capacitance, and gain of the CS stage, respectively. Notably, the second capacitance term of the Miller capacitance has a negative value when the gain of the CS stage is larger than unity. That is why this technique is referred to as the negative Miller capacitance. Accordingly, the input capacitance becomes lower than \(C_{CS}\) as long as the gain of the CS stage is larger than unity, thus improving the bandwidth of the preceding amplifier. A CTLE introduces a similar effect with the inductive feedback technique discussed in the previous section. A CTLE is generally implemented with a CS stage with source degeneration using a combination of a resistor and a capacitor. As shown in Figure 37, the source degeneration with a resistor (\(R_{S}\) in Figure 37b) decreases the gain of the CS stage by a factor of \((1+g_{m}R_{S})\) while the circuit bandwidth is extended by the same factor [107]. By placing a degeneration capacitor (\(C_{S}\)) in parallel with \(R_{S}\), the degeneration strength becomes weakened at a high frequency so that the transfer function follows that of the normal CS stage, which is similar to the effect of the inductive feedback. As a result, the CTLE achieves a high-frequency peaking in its transfer function as shown in Figure 37d. The overall photodetecting circuit presented in [106] is shown in Figure 38. Two identical CS stages are cascaded for high gain, and the negative Miller capacitance is used for the second stage to reduce the load capacitance driven by the first CS stage.