Automatic bias control suitable for a microwave photonic vector modulator

Abstract

The microwave photonic vector modulator (MPVM) sometimes shows an unmodulated carrier peak, which appears in the modulated signal. This peak results from an incorrect bias value or a Mach-Zehnder modulator (MZM) bias drift over time, which in turn leads to an imbalance in the I/Q signals. This paper describes a mathematical model of an MPVM output by showing the unmodulated carrier peak behavior. It also discusses a constellation imbalance resulting from this non-modulated peak. To control this non-modulated carrier peak and restore the constellation balance, an automatic bias control (ABC) is proposed and simulated through a co-simulation technique. Unlike previous methods, this ABC is based on the analysis of the I/Q signals to generate voltage steps for the photonic modulator bias. The I/Q signals were recovered through a direct-conversion receiver, and an algorithm was developed to analyze the I/Q amplitude and phase imbalance. Consequently, the photonic modulator bias is continuously monitored by analyzing the resulting modulated microwave signal. In this way, its operation is transparent under MZM bias variations or extinction ratio (ER) imperfection since only the I/Q and microwave signals are considered in the algorithm implementation. ABC is validated through a co-simulation technique using low and high-order modulation formats such as QPSK, 64QAM, and 256QAM.

Appendix A DPMZM derivation

The dual-parallel MZM has two MZMs embedded in an outer MZM responsible for setting a phase shift between the inner MZMs. According to Korotky and De Ridder (1990), in a square-law detector, the electrical current produced is proportional to the output intensity, which is determined by:

$$\begin{aligned} \begin{aligned} I = E^2&= E_1^2 + E_2^2 + 2|E_1| |E_2| \cos (\phi _1 - \phi _2) \\&= I_1^2 + I_2^2 + 2 \sqrt{I_1 I_2} \cos (\phi _1 - \phi _2) \end{aligned} \end{aligned}$$

(A1)

where $E_1$ and $E_2$ are the field amplitude of the inner MZMs, $(\phi _1 - \phi _2)$ is the phase shift between the inner MZMs set by the outer MZM, and $E = \sqrt{I}$ gives the optical intensity relation to the field amplitude.

Assuming that the inner MZMs are single drives in a Push-Pull configuration, the applied voltages are equal in magnitude but with opposite phases. Hence, using the trigonometric relation of $\cos ^2 \theta = \frac{1}{2} (1+ \cos 2\theta )$, and assuming a fixed optical phase shift, $\phi _0 = 90^{\circ }$, between the MZM arms, the MZM transfer function can be written as Cox (2006); Jemison and Paolella (2007):

$$\begin{aligned} \begin{aligned} I = i_{RF}(t)&= \Re P t_{ff} \cos ^2\left( \frac{\pi }{2} \frac{V(t)}{V_{\pi }} \right) \\&= \Re \frac{P}{2} t_{ff} \left( 1 + cos \left( \phi _0 - \pi \frac{V(t)}{V_{\pi }} \right) \right) \\&= \Re \frac{P}{2} t_{ff} \left( 1 + sin \left( \pi \frac{V(t)}{V_{\pi }} \right) \right) \end{aligned} \end{aligned}$$

(A2)

where $t_{ff}$ is the MZM insertion loss, P is the the laser optical power, V(t) is the applied voltage on the MZM, and $V_{\pi }$ is the half-wave voltage of MZM.

Plugging the obtained Equation A2 in Equation A1, it is possible to get the DPMZM transfer function denoted by:

$$\begin{aligned} \begin{aligned} i_{RF}(t)&=\Re \frac{P_{0}}{4} t_{ff0} \left[ \cos ^2\left( \frac{\pi }{2} \frac{V_{1}(t)}{V_{\pi 1}} \right) {+} \cos ^2\left( \frac{\pi }{2} \frac{V_{2}(t)}{V_{\pi 2}} \right) {+} 2\cos \left( \frac{\pi }{2} \frac{V_{1}(t)}{V_{\pi 1}} \right) \right. \\&\quad \left. {\times } \cos \left( \frac{\pi }{2} \frac{V_{2}(t)}{V_{\pi 2}} \right) {\times } \cos \left( \pi \frac{V_{3}}{V_{\pi 3}} \right) \right] \\&= \Re \frac{P_{0}}{8} t_{ff0} \left[ 2{+} \sin \left( \pi \frac{V_{1}(t)}{V_{\pi 1}} \right) {+} \sin \left( \pi \frac{V_{2}(t)}{V_{\pi 2}} \right) {+} 4\cos \left( \frac{\pi }{2} \frac{V_{1}(t)}{V_{\pi 1}} \right) \right. \\&\quad \left. \times \cos \left( \frac{\pi }{2} \frac{V_{2}(t)}{V_{\pi 2}} \right) \times \cos \left( \pi \frac{V_{3}}{V_{\pi 3}} \right) \right] \end{aligned} \end{aligned}$$

(A3)

where $P_{0}$ is the single laser input power, and $t_{ff0}$ is the insertion loss of DPMZM, and $V_{3}$ and $V_{\pi 3}$ are the applied voltage and the half-wave voltage of the outer MZM.

In the first line of Equation A3, the input power $P_0$ is divided by four due to a 3 dB loss of the splitter and coupler of outer MZM.