## 1 Introduction  
Typical modulation formats for free space optical (FSO) communications, such as satellite-to-ground communications, are on-off keying (OOK), differential phase-shift keying (DPSK), and polarization shift keying (POLSK) that use a direct detection (DD) receiver, or binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK) using a coherent receiver [1; 2; 3; 4]. Although the OOK, DPSK and POLSK modulations with DD receivers are inexpensive, the bit/symbol efficiencies of these modulation techniques are quite low. This problem is solved using coherent detection in which both amplitude and phase of the optical signal can be extracted [5; 6]. Among various detection approaches, coherent detection provides the best sensitivity and high spectral efficiency. However, coherent receivers are expensive, they use four photodetectors (PDs) per polarization and they require a laser which acts as the local oscillator (LO). In contrast, DD receivers require only one PD and do not need a LO. Besides, for phase-modulated signals, laser phase noise is a serious problem and, therefore, expensive external cavity lasers (ECL) with very small linewidths should be used at both the transmitter and at the receiver for coherent detection systems. In this paper, we propose the Kramers-Kronig (KK) transceiver for a long-haul FSO communication scheme in the presence of turbulence. KK receivers make use of inexpensive DD receivers. The signal phase can be extracted from the photocurrent of the PD using a Hilbert transform in digital domain. Therefore, phase-modulated signals, such as quadrature amplitude modulation (QAM), can be detected using KK relations even though a DD receiver is used. Unlike coherent receivers, a KK receiver does not require a laser at the receiver and a communication system based on a KK transceiver is not sensitive to the transmitter laser phase noise due to the use of direct detection. Therefore, inexpensive lasers with larger linewidths can be used at the transmitter. Also, digital phase noise compensation techniques are not required for the FSO system based on KK transceivers. The transmitter of a KK transceiver consists of a single Mach-Zehnder modulator (MZM) since the driving voltage is real and positive while that of a coherent transceiver requires a more complex optical in-phase and quadrature (I-Q) modulator consisting of two MZMs. Thus, KK transceivers enable high spectral efficiency communications at a lower cost. However, these benefits come with a cost--to retrieve the signal phase at the receiver, we need to add a direct current (DC) component to the signal at the transmitter which lowers the power efficiency of the transmitter, just as in intensity modulated direct detection (IMDD) systems. The KK relations are widely used in different areas of physics and engineering [7; 8; 9; 10; 11]. In [12; 13; 14], KK relations are used to design the DD coherent receivers for fiber optic communications which makes use of minimum phase signals. Recently, KK receivers have been experimentally implemented for FSO systems for short ranges of 55 m [15] and 20 m [16]. At these short ranges, the effect of atmospheric turbulence is negligible. However, to our knowledge, the performance of KK receivers in long-haul FSO communication systems in the presence of atmospheric turbulence has not been evaluated. In this paper, we study KK transceivers for FSO systems with QAM-16, 40 Gb/s data rate, and 35,000 km reach for three different scenarios: (i) Geosynchronous equatorial orbit (GEO) satellite to GEO satellite (sat-to-sat); (ii) GEO satellite to ground station (downlink); and (iii) ground station to GEO satellite (uplink), as shown in Figure 1. Besides fading, atmospheric turbulence creates extra laser beam divergence in uplink and downlink (see Figure 1) that can lead to extra path loss. We compared the proposed FSO KK scheme with a FSO coherent transceiver for downlink and uplink scenarios over weak, moderate, and strong atmospheric turbulence.