Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.

The performance of intensity modulation (IM) with direct detection (DD) transmission systems is enhanced through a novel combination of multidimensional coding, Nyquist pulse shaping, and electronic dispersion compensation (EDC) at the transmitter using a finite impulse response (FIR) filter. A 24-dimensional (24D) extended Golay binary code effectively transforms each incoming 12-bit message into a 24-bit codeword, achieving a coding efficiency of 0.5 bits per symbol for a 56 Gb/s on-off keying (OOK) transmission over 80 km of single mode fiber. While this encoding process introduces a 50% overhead, the required bandwidth is maintained at 56 GHz through doubling the symbol rate and the application of Nyquist pulse shaping with a raised cosine (RC) profile and a roll-off factor of zero, resulting in a flat power spectral density. This flat distribution contrasts with standard OOK transmission at 56 Gb/s with a roll-off factor of 1.0, where signal power is predominantly concentrated in the lower frequency range. One of the key advantages of the 24D Golay code is its substantial error correction capability. However, the benefits of this multidimensional coding and Nyquist pulse shaping extend beyond error correction. It is shown that, while both the proposed and standard OOK methods exhibit comparable performance in a white Gaussian noise channel at back-to-back, they differ significantly under frequency selective power fading conditions caused by the interplay of chromatic dispersion (CD) and direct detection. The misalignment between the frequency notches introduced by the FIR pre-EDC and those inherent in the channel response, especially severe at lower frequencies, favors transmission schemes with a flat power spectral density, like the 24D Golay-coded Nyquist pulses.

We introduce an equivalence transformation to study the oscillation behavior of solutions for linear neutral differential equations of canonical and noncanonical types. The new approach leads to several novel oscillation criteria. Moreover, we show that the same arguments can be applied to nonlinear neutral equations under suitable monotonicity conditions. The importance of the results is also supported by examples.

In this study, we investigate the oscillation behavior of second-order self-adjoint $ q $-difference equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the $ q $-version of Leighton's classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research.