## 2 Background on Atmospheric Turbulence for FSO System  
In a FSO system, the channel capacity is limited by atmospheric turbulence. Turbulence leads to random variations in refractive index which causes random changes in the amplitude and phase of the received optical signal. The fluctuation of the received signal intensity resulting from turbulence leads to performance degradation and limits the capacity. A statistical model for the optical intensity fluctuation at the receiver due to the atmospheric turbulence was derived in ref. [17] and the beam width is maximized to optimize the channel capacity. The impact of turbulence on the bit error rate (BER) was considered in refs. [18; 19].  
One of the major challenges for using the coherent detection in FSO communication systems in the presence of atmospheric turbulence is the poor mode mixing efficiency [20; 21; 22]. The spatial part of the optical field in the presence of atmospheric turbulence can be expressed as the superposition of Laguerre-Gaussian (LG) modes. Even if the optical field at the transmitter is Gaussian (i.e., LG\({}_{00}\) mode), after propagating through the turbulent medium, its energy is transferred to higher order LG modes due to the random refractive index of the turbulent medium. Typically, the spatial part of the local oscillator (LO) of the coherent receiver is Gaussian and the higher order modes of the received field do not efficiently mix with the Gaussian field of the LO due to mode orthogonality. Hence, data-LO mixing efficiency can be degraded by >20 dB due to mode mismatch between the LO and data beams [20; 21; 22]. However, an IMDD FSO link with free-space coupled photo-detector (PD) is not significantly affected by turbulence-induced modal coupling if the receiver aperture can collect most of the received field since the IMDD system does not require LO. Ref. [20] proposed a pilot assisted self-coherent detection to solve the mode mismatch problem. In this scheme [20], a pilot beam is transmitted along with the data beam from the transmitter. The pilot and data beams are more likely to undergo the similar mode-coupling to higher order LG modes due to atmospheric turbulence. The received pilot beam replaces the LO of coherent detection. However, mode mixing efficiency of the self-coherent detection is much higher than that of coherent detection. Experimental results of ref. [20] shows an average mixing loss of ~3.3 dB for self-coherent detection. The proposed KK scheme for FSO in the presence of turbulence has some similarities and differences with self-coherent detection: (i) both schemes use a pilot beam--the pilot beam of the self-coherent scheme has a frequency offset from the data beam whereas the pilot beam of the KK scheme is in-band; (ii) in KK scheme, the phase is extracted using the Hilbert transform whereas in self-coherent detection, the beating between the pilot and data is used for the data retrieval; (iii) mode mixing efficiency of the KK scheme is similar or slightly higher than that of self-coherent detection since the pilot is in-band in the KK scheme; and (iv) both schemes suffer from poor power efficiency as compared to coherent detection since a fraction of the transmitter power is wasted in sending the pilot signal.