Phase Retrieval in Short-Range Optical Communication Using Temporal Transport-of-Intensity Equation with Direct Detection

Direct detection (DD) is a straightforward, cost-effective receiving scheme for medium- and short-range fiber-optic communication systems, yet directly accessing phase information presents inherent challenges. The temporal transport-of-intensity equation (T-TIE) enables phase recovery from intensity data, but the accuracy of this phase-retrieval method is constrained by finite difference approximation errors of intensity derivatives and electrical noise interference. In this paper, we propose a 4th-order central difference method for calculating intensity derivatives to enhance approximation accuracy and implement multiple intensity measurements to further mitigate electrical noise interference. The proposed method is validated in a 28 GBaud single-carrier 16-quadrature amplitude modulation (16QAM) direct detection system. The research results indicate that, under conditions of 10 nA dark current and 20 pA/Hz^1/2 thermal noise, our method achieves a receiver sensitivity gain of 14.85 dB compared with the 1st-order forward difference method and 8.47 dB compared with the 2nd-order central difference method at the 7% hard decision forward error correction (7% HD-FEC) threshold.